Rectifiable spaces are a class of metric measure spaces that are Lipschitz analogues of differentiable manifolds (for example, they admit a parameterization by Lipschitz charts) and arise naturally in many areas of analysis and geometry. Due to the important works of Federer, Mattila, Preiss, and many others, we now have a good understanding of the geometric properties of rectifiability in Euclidean spaces. In this talk, we examine some generalizations of rectifiability to the setting of non-Euclidean spaces and discuss the similarities and differences between rectifiability in the Euclidean setting and these generalizations.