Hilbert’s Fourteenth Problem asks if certain k-algebras are finitely generated. We will look at this problem from the point of view of Geometric Invariant Theory: Say a group G acts on an affine variety X. Then we can ask whether the subring of G-invariants of the coordinate ring k[X] is finitely generated. Hilbert showed that this happens when G is reductive. However, Nagata discovered a counterexample for a choice of non-reductive G, which Mukai later  generalised. We will go over this construction of Mukai’s. If time permits, we will mention a more recent result of Bérczi, Doran and Kirwan regarding the finite generation of the subring of invariants when G is not assumed to be reductive.