What items MUST appear on a mathematician's CV? Writing a CV makes me paranoid that I'm failing to abide by unwritten rules. Of course CVs are flexible to capture the diversity of accomplishments someone might have. But there must be plenty of things a hiring committee absolutely expects. So I'm interested in anything that must be on a CV — whose omission would raise a red flag — of a mathematician looking for an academic job.
"Obvious" answers are welcome. Even things which sound obvious like "your name." What is obvious to someone who has read and evaluated lots of CVs is different from someone preparing one for the first time.  In your answer, please also be fairly specific about scope: have you served on hiring committees? for what types of positions? in the US or Europe or?
(Note: As suggested in the comments, it is very good to ask people "in the know" directly for such career advice.  One reason for asking this question on MO is to have more open, less clubby answers — there is an echo chamber effect when you ask a bunch of people in the same subcommunity.)
 A: From my perspective, the critical question isn't what must be included on your CV, but rather what mustn't, since that seems to be the more common problem (judging by the ones I see). What I'm about to describe is based on my experience at a U.S. research lab; I imagine it generalizes quite a bit beyond that, but I can't say how far, and it is certainly country-specific. 
I'll discuss five rules below, with some overlap between them. Of course these rules are not absolute (except for the last one), but you certainly shouldn't break them without thinking carefully about it and deciding there's a good reason to do so.
(1) Your CV should represent you as a professional mathematician. Anything that is not relevant to your professional life should be left out. For example, you should generally not describe non-math-related summer or part-time jobs, hobbies, side interests outside of mathematics and related fields, etc. If there's something unusually interesting or impressive (you published a novel or are a chess champion) or that displays relevant skills (you write free software in your spare time), it's OK to mention it, but just briefly and not in a prominent position.
I've seen some hair-raising violations of this rule, in which applicants devoted considerable space to things that have nothing to do with working as a mathematician. Nobody is going to reject your application just because you put something weird in your CV, but it's not good for your image as a professional.
(2) Your CV shouldn't include anything unless you think the search committee might need or want to know it. For example, contact information is valuable, as is anything that can legitimately help judge your application. However, in the U.S. you should not list your age or birthdate, your marital status, information about your children, or your religion (unless you are applying to a religious institution). I realize this is common in some countries, and of course people will be understanding about that, but it comes across strangely to give people information they don't want and shouldn't be influenced by.
(3) You should try not to seem desperate to impress, particularly with awards and distinctions. Some people provide enormous lists of very minor distinctions, sometimes with no relevance to research/teaching/service (for example, a college scholarship from a local business club). Coming across as insecure can make you seem less attractive: an ambitious department wants to hire people who are marginally too good for them, not people who are trying hard to be good enough. As a rule of thumb, when you get your Ph.D. and apply for your first job, it's OK to list any substantive distinction from grad school. You can list a few undergraduate honors, but only if they are impressive (Putnam fellow or major university-wide prize, yes; random scholarship, no). You shouldn't list high school honors at all (well, just maybe an IMO medal, but be careful not to look like you consider it your proudest achievement).
(4) Be sure not to give the impression you are trying to obfuscate anything. I don't just mean you should tell the truth, but also that you should be clear and straightforward. For example, people sometimes feel bad about not having enough items to list in their publication or talk sections, and it can be tempting to reorganize the CV to try to obscure this. For example, you could replace the "publications" section with a "research" section in which you list not just publications but also talks and poster presentations, or even current/future research topics. This is a bad idea, since it can look like you are trying to make the information less accessible, and then everything on your CV will be looked at more skeptically. Instead, you want to make it easy to understand your CV and easy to see that you aren't doing anything tricky.
(5) Don't lie. Don't say a paper will appear in a journal until it has been accepted, even if you are sure it will be. Don't say a paper is submitted until it is, even if you plan to submit it by the time the committee meets. Don't call something a preprint until it is written down and ready to distribute (you can say "in preparation" before then, but many people will ignore this since it is unverifiable). Don't say you have received a fellowship or prize if you haven't. You'd think all these things go without saying, but I've seen a couple of people get caught on one of them. You really don't want to be the person who gets asked for a copy of their preprint and can't produce one.
A: *

*Include the URL of your web page (and have one!).

*Include the arXiv references of your papers, and in the PDF version make those arXiv references actual links to the papers (e.g. using the hyperref package).

A: An obvious answer
One cannot insist enough (and I am surprised to see this has been hinted at already, but not stated quite this boldly) on the fact that different positions have different expectations, e.g. many non-academic employers seem to expect a single-page résumé.
Even when looking for an academic position, candidates routinely keep more than one version of their CV. In order to tailor to the specifics of the various positions (more or less emphasis on research, post-doc vs. tenure-track). Even if you put the exact same items in all your CVs, the order in which they are presented, which items are emphasized and detailed is a good way prove that you understand the expectations of the position, and make sure that the relevant items are easily found by the reader.
I cannot be more specific since this question yet again commits the sin of being a non-geographically specific career question: needless to say that localization also plays a role.
A: I second the advice about modeling your CV on those of others. Personally, when I'm reading CV's of job applicants, here is what I first look at:


*

*Name, employment history, education (the latter two switched in order of importance if the person is a very recent PhD)

*research interests

*publications

*honors, awards, editorial work


At a second pass, I'd look at


*

*talks

*teaching

*PhDs/postdocs supervised (depending on the position advertised)


Unless these are explicit requirements of the position (senior hires, hires to administration), I find information on grants and department-level service not very helpful. Granting systems in different countries vary wildly, as does the nature of what's service. 
Caveat emptor: this is how I read CVs, and is not intended to imply anything of a universal nature.
(added later to provide scope): I've served on hiring committees in Canada for postdoc, junior and senior faculty searches (open and targetted), university senior administrators, prize committees (for research awards) and for granting agencies in North America and Europe. So my experience is limited.
A: As Henry suggests, you should just model it on other people's CVs (you can use mine; I ripped it off from Ezra Miller), and ask a more senior person if you're uncertain about an item.  The obvious items are (it's CW, so people can add):
On practically every CV


*

*Name

*Contact Info

*Employment History

*Education

*Grants, Honors, etc.

*Publications

*Service

*Teaching


Things you might include (but which I don't necessarily recommend):


*

*Citations

*Talks

*Programming languages or foreign languages spoken

*Names of people who will vouch for you

A: Your PhD advisor's name (listed under education).
A: I will second Thierry's reply.  Of course, when I see the term "CV" I think of academic positions, whereas if one is seeking a job in industry one would instead produce a "resume".  Different beasts, though in both cases the purpose is to some extent the same, namely to give an employer some idea why they should be interested in you to fill their opening.  As Thierry said, a hiring manager in a corporation or industrial firm will have a very different set of things he is looking for in a resume compared to what a hiring committee in a math department at some college or university expects in a CV.
