It is well known that if G is a finite group then the group of endotrivial modules is finitely generated. In this paper we investigate endotrivial modules over arbitrary finite group schemes. Our results can be applied to computing the endotrivial group for several classes of infinitesimal group schemes which include the Frobenius kernels of parabolic subgroups, and their unipotent radicals(for reductive algebraic groups). For G reductive, we also present a classification of simple, induced/Weyl and tilting modules (G-modules) which are endotrivial over the Frobenius kernel G_r of G.