Are there very famous open problems or conjectures in representation theory, or in enumerative geometry, like the volume conjecture in topology?
The Clemens conjecture in enumerative geometry: a general quintic threefold has only finitely many rational curves in each positive degree.
There are many open, and seemingly deep, conjectures in modular representation theory (or block theory) in connection with enumerating representation-theoretic invariants: a start of a list might be : Brauer's $k(B)$-problem, the Alperin-McKay Conjecture, the Alperin Weight Conjecture, Dade's conjectures, Isaacs-Navarro conjecture. Gabriel Navarro has several recent survey papers discussing these and other conjectures.
In a different part of (modular) representation theory, with perhaps a more geometric flavor, there are problems such as the Lusztig Conjecture (now known to be false in its original formulation), and work of Geordie Williamson.
As noted by Julian Kuelshammer, Representation Theory is a vast subject, and it might be helpful to point out which specific areas you are most interested in (I only mention two facets of the subject which are most familiar to me).