  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.
