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1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all. [But see my acquiescence to Jukka Suomela's compelling point in the comments.]

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the perceived quality of the paper. I have spent upwards of 100 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct).

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently advanced in the academic hierarchy to render this without risk.

1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all. [But see my acquiescence to Jukka Suomela's compelling point in the comments.]

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the perceived quality of the paper. I have spent upwards of 50 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct). [I originally said 100 hrs, but upon reflection I think it only felt like 100 hrs.]

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently senior to render this without risk.

1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all.

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the perceived quality of the paper. I have spent upwards of 100 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct).

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently advanced in the academic hierarchy to render this without risk.

1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all. [But see my acquiescence to Jukka Suomela's compelling point in the comments.]

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the perceived quality of the paper. I have spent upwards of 100 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct).

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently advanced in the academic hierarchy to render this without risk.

1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all.

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the quality. I have spent upwards of 100 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct).

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently advanced in the academic hierarchy to render this without risk.

1. Should you summarize the main results and or the argument?

In general, I would say: No. However, for some journals I have come to know the overwhelmed editors who in fact need this summary. So: Write it for the editors, if you write it at all.

2. What do you do when the journal requests you to evaluate the quality of the result (or its appropriateness for the journal)? Should one always make a recommendation regarding publication?

This requires you to know the standards of the journal in question. If you do, then definitely evaluate according to those standards. For example, I review for Discrete & Computational Geometry and for Geometriae Dedicata, which both have high standards and long backlogs. I take that into account. Does this paper deserved to be published in the all-too-few pages that these prestigious journals can provide each issue, rising above many others?

3. What to do about papers with major grammatical or syntactical errors, but that are otherwise correct? Does it matter if the author is clearly not a native English speaker?

I view my job as referee to evaluate both the quality of the mathematics and the comprehensibility of the writing. I sometimes will edit just one paragraph for style and clarity (with maybe ten what I think are improvements), and suggest that this be done throughout. But it is not the job of a referee to be a style-editor, nor a 2nd-language improver.

4. On this note, at [what] point does one correct such mistakes? Ever? If there are fewer than a dozen? Should one be proof-reading the paper?

I may be abnormal in this respect, but my referee reports are not uncommonly $\ge5$ pages in LaTeX.

5. What do you do when you do not understand an argument? Does it matter if it "feels" correct. How long should one spend trying to understand an argument?

Yes, it matters! Say it "feels correct" but you have not verified it in detail. The question of "how long" depends very much on the perceived quality of the paper. I have spent upwards of 100 hrs on what I thought was a seminal paper, but have dispensed with equally long papers in an hour or two if I thought their results were pedestrian (even if correct).

6. What to do about papers that have no errors but whose exposition is hard to follow?

You say exactly that, detailing where and why. How to proceed from here is an editorial decision, not a referee decision.

Finally, I will say something perhaps controversial, but not apropos RBega's question. I think anonymous refereeing is funadmentally defective in the abuses it invites, and therefore I only review nonanonymously. But I am sufficiently advanced in the academic hierarchy to render this without risk.