The aim of this thesis is to study the action of a finite group G on a polynomial ring over a principal ideal domain (PID). We focus on finding the ring of invariants under this action and study its properties. We transfer many of the fundamental results from invariant rings over fields to invariant rings over PID's, in particular over the ring of integers. We find the generating sets for invariant rings Z[Lp-1]Cp and Z[Lp-1 + Lp-1]Cp as Z-algebras, where Lp-1 is the indecomposable ZG-module of rank p-1. We also study some properties of the ring of invariants such as the Cohen-Macaulay and the Gorenstein property. We show certain results over fields hold over the ring of integers. We also find a degree bound for the generators of the invariant ring over the integers in many cases and compare this bound with the bound that holds over the rationals. In addition, we give some counterexamples concerning Noether's bound and the Cohen-Macaulay property. We provide a complete list of generating sets for the invariant ring over Z in two and three variables under the action of a finite group G. These generating sets are given by providing primary and secondary invariants.