A question about MathSciNet etiquette Hello,
Recently, a colleague of mine pointed me to a MathSciNet review of one of my papers that is completely off the mark - it is not negative or anything like that, but it grossly misrepresents the contents of the paper (when describing the origins of the techniques and questions in the paper, for instance, as well as the "position" of the results within the current litterature and the meaning of the results themselves).
I'm not sure what I should do - actually, I probably won't do anything because the paper seems pretty inconsequential and the quality of the review most likely does not matter much. Still, the same thing could have happened with a paper I truly care about, and this led me to wondering what the proper behaviour is: should I contact the reviewer and ask him/her to retract his/her review (explaining why, of course)? Should I contact MathSciNet and let them know that I believe the review is incorrect? The first option raises some "diplomatic" problems, while the second one seems to me both to be abrupt and to waste several people's time... I think, if pressed to act, I would choose the first course of action, but I'd be grateful for any suggestions (e.g on how to say "you absolutely mangled that review!" without being rude..)
Final note: I'm asking this question anonymously because I don't have that many papers and it would be easy to identify the reviewer by looking at my MathSciNet profile, and I'm not out to embarass anyone.
Thanks for your help - if the question is inappropriate for this site then please close, of course!
 A: It is in principle possible that reviews are revised/replaced (as I assume you know).
However, as far as I know, this is mainly done to replace factual errors. From your description it seems to me that your situation might be at the borderline of factual errors and different interpretation. 
You could also check http://www.ams.org/mresubs/guide-reviewers.html (in particular the point 'evaluative reviews' to see whether the review is in line with what is written there).
Yet, since you say that the review is not negative, this seems less relevant to your situation.
Now, in case you decide to do anything, in your situation it seems definitely advisible to contact the reviewer directly. As it was not negative it might be a misunderstanding/an honest mistake, and the reviewer might be willing or in some sense even happy to correct the review if made aware of the problem. 
I am a fairly frequent reviewer for MathSciNet. My greatest worry doing this is to write something wrong in a review, and by doing so, somehow make a fool out of myself. So, in case an author of a paper I reviewed should ever contact me in a friendly way and point out why my review is not to the point and I then understand that s/he is right, I certainly would try to get this corrected, and be grateful for being made aware of it.  
By contrast, I know indirectly (a collegue of a collegue...you get the idea) that if the reviewer does not want to change the review it can be very difficult to get it changed (except there is a direct factual error).
So, you need the reviewers consent anyway, and as André Henriques says, there is no way to hide that you were 'behind' the activity; or even if there was a way to do this anonymously, it should be the default assumption that the author of the paper is the one 'complaining'.  This seems another reason for directly contacting the reviewer, as it seems rather more likely to get her/his consent by this course of action, then by one going an official way. 
How to phrase the communication with the reviewer: a way could be to not directly or mainly critizise the review, but to just sent the reviewer your point of view. In other words, write somehow your own review and share it with the reviewer. Perhaps only pointing out key differences in passing. 
In any case, if you do not care deeply I would at most contact the author of the review; perhaps this works out well, but if not rather drop the matter as further activity might be in vain, and not be worth the effort and negative side-effects. 
A: Like quid, I'm a regular reviewer, with about 60 reviews to date. The large majority of those reviews have been untouched since they first appeared, but in a few cases I have later asked Math Reviews to implement light edits to correct errors. The edited reviews appeared in due course. (That is, they appeared in MathSciNet; I don't know the policy about the print version.) On one occasion, I did this because the author contacted me to point out a mistake in my review, which I was glad to correct. 
In writing a review, I'm aiming for concision. But I try to touch on 1) the context of the work, 2) the main results, and 3) what sorts of ideas go into the proofs. I read only in as much detail as I need to do that. Occasionally, when I feel that I have a distinctive perspective on material in the paper I'm reviewing, or its relation to other papers, I comment on that perspective. I try to do so sensitively and sparingly: I am conscious that these comments should add to the information I convey about the article, not replace it. But I would argue that as a reviewer it is my privilege to explain the context of the work as I see it.
If the problem could be resolved by straightforward corrections to the review, I would suggest that you contact the reviewer directly. If the review is so misleading as to make you think that the reviewer shouldn't be writing about papers in that area, you could instead make that case to Math Reviews - and perhaps enlist your expert colleague to do the same.
A: Thanks to everyone for the advice. It seems that the question has stopped attracting new reactions, so I'll try to summarize what I took from the discussion.


*

*There is no way (nor should there be) or hiding the fact that I'm the one complaining about the review's quality - anyway, the reviewer signed his/her review so it would be highly discourteous to try to stay anonymous.

*In all probability the reviewer tried to do a fair job and simply failed; it does not seem unlikely that he/she would prefer to be told that the review is off the mark and have a chance to make it right. This helps prevent his/her name to be attached to an erroneous review. 

*In view of all this, it seems at least courteous to begin by contacting the reviewer and explaining why the review seems wrong to me, and then give him/her a chance to correct it on his/her own (or to explain me why I misunderstood my paper, of course). 

*In the course of such communication, one should as much as possible avoid being too directly critical of the original review (and remember that the reviewer wrote it on what is essentially his/her own free time!).

*If the above option does not produce the desired results, then the way to go (if sufficiently motivated!) is to contact mathrev@ams.org, explain the issues with the review, and leave it to them (at least one contributor thinks that this should be the first step, without any direct contact with the reviewer, but I do not see the downside of contacting the reviewer first and MathSciNet second).


Also, note that a somewhat similar question may be found at How do I fix someone's published error? , and that some of the advice there is relevant to the problem at hand.
Again, thanks to everyone for the advice! 
Note possibly to be removed at a later point: I plan to accept this answer, so that it appears first to anyone interested in a similar problem at a latter date. I'll wait a few days before doing so, that way you can let me know if this clashes with this forum's usual rules.
A: The correct method is to write to mathrev@ams.org, and describe your concern in detail, and leave the communication with the reviewer with them.
