William Thurston's seminal construction of a hyperbolic 3-manifold fibering over the circle gave the first example of a Gromov hyperbolic surface-by-cyclic group. This breakthrough sparked a flurry of activity, and there has subsequently been much progress towards developing a general theory of hyperbolic group extensions. In this talk I will review some of this basic theory -- including combination theorems for ensuring a group extension is hyperbolic and structural theorems about general hyperbolic extensions -- and then discuss my work with Sam Taylor studying hyperbolicity in the specific context of free group extensions. For instance, we use the geometry of Outer space to show that every purely atoroidal subgroup of Out(F_n) that quasi-isometrically embeds into the free factor complex gives rise to a hyperbolic extension of F_n.