We provide an introduction to the cohomology of finite group schemes, a of objects which includes finite groups and $p$-restricted Lie algebras. Various qualitative results, known earlier for finite groups by work of Quillen and others, are extended to this general context. Various computational techniques which arise from ical homological algebra are recalled. We then proceed to discuss the essential role of strict polynomial functors in the proof of the fundamental theorem which asserts that the cohomology of a finite group scheme is finitely generated.