This thesis explores the space of flat metrics sitting in the space of geodesic cur- rents. The space of geodesic currents, considered as the completion of weighted closed curves, was first introduced by Bonahon to study hyperbolic 3–manifolds, and then used by him to provide an alternative description of Thurston’s compactifica- tion of Teichmu ̈ller space. This space has since played a key role in the study of geometric structures on surfaces and their degenerations. We focus on the space of flat metrics from k–differentials on finite–type surfaces and its embedding into the space of geodesic currents. With this embedding, we describe a compactification of the space of flat metrics coming from k–differentials, for all k, where the boundary points are the mixed structures—currents that are a flat metric on a subsurface and a measured lamination on the complement. This generalizes the results of Duchin– Leininger–Rafi for the case when k = 2 and Ouyang–Tamburelli for the case when k = 3, 4.