For topological spaces with an action of a group G, an isovariant map is an equivariant map which preserves isotropy subgroups. Isovariant maps play an important role in equivariant surgery theory and equivariant h-cobordism theory. In this talk, I will discuss joint work with Sarah Yeakel, in which we study the homotopy theory of G-spaces with isovariant maps. Results discussed may include an isovariant Whitehead's theorem, an application to isovariant fixed point theory, and the beginnings of isovariant stable homotopy theory.