Not to sound like a refusenik or a contrarian, but I have always been a bit of an agnostic when it comes to the arXiv. Somewhere deep in my old Pine folders there is a 15 y.o. discussion with Greg Kuperberg on the subject, and it seems the history proved him right - the arXiv is now incredibly valuable and popular (perhaps, even a little too popular, see below). However, despite all the benefits, I think posting on the arXiv is a serious decision, often enough a good idea, but not always, and definitely not without downside as Matthew Daws writes (to a wide 27+ support):

1. If you are a graduate student or a junior faculty, you might not be as fast as others in developing your own ideas. As soon as you post early results which use your new ideas, they become a fair game. Now someone else can recognize their value and quickly solve your main problem before you are half way through. On the other hand, going the more traditional print publication route would give a couple of years cushion, sufficient in most cases.

2. On a related subject of destructive competition, arXiv can be really unhelpful. One graduate student I know liked a conjecture posed in an arXiv preprint. He solved it in about two months. When he was finishing writing his solution, somebody posted an identical solution. He was quite distressed. Another graduate student I know, discovered two weeks before the Ph.D. defense that the main result in her thesis was just posted on the arXiv in a more general form by a senior faculty elsewhere. Upon insistence of her advisor, she cancelled the defense and left academia without a Ph.D. degree.

3. An obvious point: some/many arXiv preprints are incorrect, leading to questions like this one. This creates a bizarre "neither solved nor open" status: while a solution of an important problem is being checked, no one wants to work on the problem. On rare occasions, two opposite solutions are posted leading to partial paralysis in the field. BTW, plagiarism is yet another variation on this issue (here the authorship is incorrect).

4. The other side of the same coin: if a person (like one friend of mine) posts an incorrect solution of a famous problem, this creates too much attention, potentially destroying a career (esp if in the early stages).

5. The "unaffiliated people problem" which makes it hard from people from third world countries, as well as anonymous authors to contribute (not everyone is as brave as Mnёv, see my answer here). While arXiv's restrictions do help get rid of some cranks, there are other ways to do that, and one can argue that one gem from an unaffiliated author is worth 100 crank papers.

6. Some people apparently read arXiv every day. Really? One friend of mine (in physics) admitted to me he spends 1.5-2 hours every morning doing that. Really???

7. ArXiv's success also has downsides, as it increases pressure on young mathematicians to post, so as to keep up with others, even if their natural instinct in some cases maybe is to be protective of their ideas and further develop them before making them public. The social pressure can be quite strong, as hiring committees increasingly view arXiv preprints as "near publications". More anecdotally, one senior mathematician I know likes to ask people "What's your most recent arXiv paper?" in place of a more traditional "What are you working on?" as if arXiv posting is a "must do" for everyone.

8. An indirect and less obvious downside: arXiv's success clearly slowed down rather than sped up the natural tendency to bypass traditional print publishers for electronic media, as there is less of a pressure to have all journals nearly free and widely accessible. This is contrary to the early predictions which expected for the transition to happen before 2010.

Not to sound like a refusenik or a contrarian, but I have always been a bit of an agnostic when it comes to the arXiv. Somewhere deep in my old Pine folders there is a 15 y.o. discussion with Greg Kuperberg on the subject, and it seems the history proved him right - the arXiv is now incredibly valuable and popular (perhaps, even a little too popular, see below). However, despite all the benefits, I think posting on the arXiv is a serious decision, often enough a good idea, but not always, and definitely not without downside as Matthew Daws writes (to a wide 27+ support):

1. If you are a graduate student or a junior faculty, you might not be as fast as others in developing your own ideas. As soon as you post early results which use your new ideas, they become a fair game. Now someone else can recognize their value and quickly solve your main problem before you are half way through. On the other hand, going the more traditional print publication route would give a couple of years cushion, sufficient in most cases.

2. On a related subject of destructive competition, arXiv can be really unhelpful. One graduate student I know liked a conjecture posed in an arXiv preprint. He solved it in about two months. When he was finishing writing his solution, somebody posted an identical solution. He was quite distressed. Another graduate student I know, discovered two weeks before the Ph.D. defense that the main result in her thesis was just posted on the arXiv in a more general form by a senior faculty elsewhere. Upon insistence of her advisor, she cancelled the defense and left academia without a Ph.D. degree.

3. An obvious point: some/many arXiv preprints are incorrect, leading to questions like this one. This creates a bizarre "neither solved nor open" status: while a solution of an important problem is being checked, no one wants to work on the problem. On rare occasions, two opposite solutions are posted leading to partial paralysis in the field. BTW, plagiarism is yet another variation on this issue (here the authorship is incorrect).

4. The other side of the same coin: if a person (like one friend of mine) posts an incorrect solution of a famous problem, this creates too much attention, potentially destroying a career (esp if in the early stages).

5. The "unaffiliated people problem" which makes it hard from people from third world countries, as well as anonymous authors to contribute (not everyone is as brave as Mnёv, see my answer here). While arXiv's restrictions do help get rid of some cranks, there are other ways to do that, and one can argue that one gem from an unaffiliated author is worth 100 crank papers.

6. Some people apparently read arXiv every day. Really? One friend of mine (in physics) admitted to me he spends 1.5-2 hours every morning doing that. Really???

7. ArXiv's success also has downsides, as it increases pressure on young mathematicians to post, so as to keep up with others, even if their natural instinct in some cases maybe is to be protective of their ideas and further develop them before making them public. The social pressure can be quite strong, as hiring committees increasingly view arXiv preprints as "near publications". More anecdotally, one senior mathematician I know likes to ask people "What's your most recent arXiv paper?" in place of a more traditional "What are you working on?" as if arXiv posting is a "must do" for everyone.

8. An indirect and less obvious downside: arXiv's success clearly slowed down rather than sped up the natural tendency to bypass traditional print publishers for electronic media, as there is less of a pressure to have all journals nearly free and widely accessible. This is contrary to the early predictions which expected for the transition to happen before 2010.