One of the major problems is whether K[V]G is always finitely generated as an algebra over K. This is known as Hilbert's 14th problem (one of the problems he proposed at the beginning of the 20th century to keep everyone busy). Before, in 1890, he proved that for G=SLn, G=GLn and other classical groups. It can be shown that K[V]G is finitely generated for any reductive group. However, Nagata found a counterexample for Hilbert's 14th problem where G is not reductive.
I am very much interested in the constructive aspect of invariant theory. One of my contributions is an algorithm for computing a finite set of generating invariants for K[V]G if G is linearly reductive. Let b(V) be the smallest integer d such that K[V]G is generated by invariants of degree less or equal than d. I also obtained some good upper bounds for b(V).
There are several reasons for studying quiver representations. To each quiver one can associate its path algebra. Any finite dimensional associative algebra is just a path algebra modulo some ideal (up to Morita equivalence). So study of finite dimensional associative algebras leads to the study of quivers. Results of Kac, Lusztig and Nakajima also link quiver representations to Kac-Moody algebras and Quantum groups etc.
I am particularly interested in the connection between quiver representations and the representation theory of GLn. Studying the behaviour of ``generic'' representations of quivers, one can obtain results about Littlewood-Richardson coefficients.
Conjecture: If XxK is isomorphic to Kn+1 for some affine variety X, then X must be isomorphic to Kn.
or this conjecture by Abhyankar and Sathaye:
Conjecture: If f:Kn--->K is a polynomial such that f-1(0) is isomorphic to Kn-1, then any fiber f-1(a) is isomorphic to Kn-1 (or even stronger: f must be a polynomial coordinate).
and then there is the linearization conjecture:
Conjecture: Let G be a linear algebraic group acting rationally on Kn. Is the action of G linearizable, i.e., is the action linear after a polynomial change of coordinates?
To this last conjecture there are counterexamples in positive characteristic (Asanuma), if G is not commutative (Schwarz, Knop and others), and in the holomorphic category (Kutzschebauch and myself). However it is still open if G is abelian and K has characteristic 0.