TITLE: how to read a mathematical paper?
QUESTION [67 upvotes]: I hope that this question is on-topic, though it is not quite technical.
I am curious to hear from people how they approach reading a mathematical paper.
I am not asking specific questions on purpose, though at first I had a few. But I want to keep it rather open-ended.

REPLY [7 votes]: I generally have a "top down approach" to reading papers. More often than not, I am driven to a particular paper looking for a particular result. So I would start by looking at the statement of the result and try to see what background is needed in order to parse the statements which appear there. Once I have understood the philosophy of the result, I think about/look at the proof. Again, if I come across something unfamiliar, I try to acquaint myself with the relevant material on an as needed basis. I find this approach works better for me if I am interested in a paper for isolated results. Reading such papers from introduction to conclusion might not be of immediate (or even future) value.
The other kinds of papers I read would include survey papers (exposition of a particular new idea) and papers which essentially develop a new idea/concept/structure for the first time. A lot of these papers, I find, can (and eventually should) be read from front matter to bibliography. So I would essentially read these papers as I would read a textbook.
One thing which I find especially important while reading papers is to look at the cross references. This can be useful for motivation, alternate proofs and related ideas. It also gets me acquainted with who else is working in the area and the exposure to ideas from different mathematicians achieved thus is a nice side effect. 
Finally, I feel it is important to ask a lot of questions while I read papers. Why is hypothesis X necessary. Can condition Y be weakened? Can this result be generalized to Z structure? This is eventually useful when I write my own results, to conceive the most general form that I can.