The fourteenth Hilbert problem

The 14th Hilbert problem is formulated as follows: is the algebra of polynomials in $n$ variables, invariant under a subgroup $G$ of $GL(n)$, finitely generated? In 1958, Masayoshi Nagata presented his famous counterexample to this problem. However, at this moment, there is no complete description of the groups $G$ for which the answer is affirmative; there are only a few constructions for counterexamples. We will survey various positive and negative results in this area.