A vortex in a straining field is a canonical situation describing vortices in an irrotational flow. Exact solutions to this problem have been found in the form of vortex patches and hollow vortices, both of which can be viewed as desingularizations of point vortices. After a review of the history of point vortices, we discuss hollow vortices, which are steady vortex sheets. We then focus on the case of vortices in strain and examine hollow vortices and vortex patches, describing the bifurcation structure of the latter. Finally we consider Sadovskii vortices, which contain both interior vorticity and a vortex sheet on the boundary, and sketch the relations between the different solutions.