The famous Positive Mass Theorem of Schoen-Yau and Witten states that an asymptotically Euclidean manifold with nonnegative scalar curvature must have nonnegative ADM mass (if the dimension of the manifold is between $3$ and $7$ or if the manifold is spin). Moreover, the mass is zero iff the manifold is the Euclidean space. Recently there has been a lot of interest and activity in Positive Mass Theorems for spaces which are asymptotically approaching an Euclidean space times a compact manifold. We will discuss some of the work in this direction going back to our earlier work as well as some of our recent work and related work by others.