In the first part of the talk I give an introduction to the computational tool called ``Hilbert Series'' (HS). I analyze how it can be employed for the characterization of the moduli space of vacua of a QFT and of the moduli space of instantons. Then, in the second part of the talk, I discuss the moduli space of (framed) self-dual instantons on $CP^2$. These are described by an ADHM-like construction which allows to compute the Hilbert Series of the moduli space. The latter has been found to be blind to certain compact directions. I probe these directions, finding them to correspond to a Grassmanian, upon considering appropriate ungaugings. Moreover I discuss the ADHM-like construction of instantons on $CP^2/Z_n$ as well as compute its Hilbert series. As in the unorbifolded case, these turn out to coincide with those for instantons on $C^2/Z_n$. \\ \\ This talk is mainly based on https://arxiv.org/abs/1502.07876 .