TITLE: Open problems/questions in representation theory and around?
QUESTION [43 upvotes]: What are open problems in representation theory?
What are the sources (books/papers/sites) discussing this?
Any kinds of problems/questions are welcome - big/small, vague/concrete.
Some estimation of difficulty and importance, as well as, small description, prerequisites and relevant references, ... are welcome.

To the best of my knowledge, there are NO good lists of representation theory problems on the web. E.g. the sites below contain lots of unsolved problem in other areas, but not in representation theory:
http://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics
http://garden.irmacs.sfu.ca/
http://maven.smith.edu/~orourke/TOPP/
MO questions also discuss other fields, but not representation theory:
What are the big problems in probability theory?
What are some open problems in algebraic geometry?
What are some open problems in toric varieties?
More open problems
Open problems with monetary rewards
Open problems in Euclidean geometry?
Open Questions in Riemannian Geometry
What are some of the big open problems in 3-manifold theory?
Open problems in continued fractions theory

REPLY [12 votes]: There are many open problems on modular representation theory of finite groups. There is a well-known list of problems of R. Brauer which date to around 1960, about ordinary and modular representations of finite groups. There are other major conjectures ( Broue's Abelian defect group conjecture has already been mentioned in comments). There are Alperin's Weight conjecture, Dade's Projective Conjecture, the Isaacs-Navarro conjecture, and several related conjectures. This is only the tip of the iceberg (and only in one corner of representation theory).