A classical example in nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it.
Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group.
Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k. Zariski's formulation was shown[1] to be equivalent to the original problem, for X normal.
(Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.[2] Nagata (1960) gave the following counterexample to Hilbert's problem.
For example, Totaro (2008) showed that over any field there is an action of the sum G3a of three copies of the additive group on k18 whose ring of invariants is not finitely generated.