Rectifiable spaces are Lipschitz analogues of differential manifolds. 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 will examine 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.