I have contacted a professor (in Europe) informally and asked for a PhD position. She answered in positive manner and asked for a CV.
What should I include? Do people in academia care about the usual 2-page restriction?
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Generally I would say No, people do not put restrictions on CVs unless specifically asking for it.
In a case such as the one you decribe you should send a complete CV that lists everything that can be meriting for the position you will apply for. The following would be of interest:
If you have written a paper (thesis) of some sort, provide a copy. Only one, the most important though. If you do not have one that is fine, no one would expect you to have written much.
Despite the length of the list try to make it brief and clear so that it can be assessed with ease. ry to find a good layout that make sthe structure easy to see.
Someone else had mentioned this on a related question here: the words "curriculum vitae" are taken from the Latin for "course of my life". So there's not much point in a page restriction :), unlike a corporate resume, which is often required to be one page or one sheet.
As for what you should put in it, Peter's answer is very comprehensive.