The essential idea for a partial frame is that it should be “frame-like” but that not all joins need exist; only certain joins have guaranteed existence and binary meets should distribute over these joins. The guaranteed joins are specified in a global way on the category of meet-semilattices by specifying what is called a selection function.In this paper, partial spaces are the primary object of interest. A partial space is a set together with a collection of its subsets that forms a partial frame under intersection and union.We consider lower and higher separation axioms; we note, for instance, that much of the classical behavior of regularity can be recovered by using a partial frame approach.Using the adjunction between partial spaces and partial frames, one can define the important class of sober partial spaces as those fixed by the adjunction. These play an important role in this study, since the classical result that Hausdorff spaces are sober does not hold in the partial setting.We construct the regular coreflection for partial frames and exploit this to provide a regular reflection for partial spaces. Similar constructions suffice to produce completely regular and zero-dimensional reflections.Finally, turning to compactness, we provide compactifications and zero-dimensional compactifications for partial spaces.