TITLE: Refereeing a Paper
QUESTION [219 upvotes]: I've refereed at least a dozen papers in my (short) career so far and I still find the process completely baffling.  I'm wondering what is actually expected and what people tend to do...
Some things I've wondered about:
1) Should you summarize the main results and or the argument?  If so, how much is a good amount. What purpose does this serve?
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? 
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?
4) On this note, at point does one correct such mistakes?  Ever?  If there are fewer than a dozen? Should one be proof-reading the paper?
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?
6) What to do about papers that have no errors but whose exposition is hard to follow?

REPLY [25 votes]: The referee must check if the result is correct! That is, the referee should check the proofs. This point seems to be so under-estimated, but in my opinion it is the most important part of the referee's job. The referee is the only one who will do this, except the author. There does need to be a second check of correctness after the author. There are many flawed papers out there. This has cost many other researchers time and health.
If the referee is unsure, he should ask the author for help in understanding the proofs (and the author should be helpful). It can be a change in the paper or a note for the referee.
As for whether or not the paper is "interesting": I decline to referee a paper if it is not interesting (in a broad sense) enough for me to put the effort in checking whether or not it is correct. Thus if all referees were correctness minded, then "interesting" papers would get their advantage.