TITLE: Classification compact Riemann Surfaces
QUESTION [7 upvotes]: I know that all compact Riemann surfaces with the same genus are topologically equivalent. Moreover they are diffeomorphic. But are they biholomorphic, too?
In other words, is the complex structure conserved?

REPLY [2 votes]: The answers here are great, but I think anyone drawn to this question should find the keyword "Teichmüller space" somewhere and maybe some references.  Here is the wikipedia page as a start.
And Chapters 10-12 in A primer on mapping class groups serves as a great introduction.  See also here for more suggestions.
Note:  with the top of page 10 (ibid) in mind, in general complex smooth varieties that are biholomorphic need not be biregular (see here).