While stable sheaves on surfaces with trivial canonical bundle, i.e. K3 or Abelian surfaces, have been extensively studied, the Enriques case is much less understand. We will first discuss the past work of Kim, Hauzer, and Yoshioka on some existence and irreducibility results, and then present our new results which give necessary and sufficient conditions for the existence of stable sheaves with prescribed chern classes. Furthermore, we will discuss some new irreducibility results as well as some of the geometry of these moduli spaces. Time permitting, we will describe a near complete on-going project describing the MMP of these moduli spaces using Bridgeland stability techniques.