In this paper we develop a theory of non-abelian cohomology for proalgebraic groups which is used in J. Amer. Math. Soc. 24 (2011), 709-769 to study the unipotent section conjecture. The non-abelian cohomology H 1nab(G,P) is a scheme. The argument G is a proalgebraic group; the coefficient group P is prounipotent with trivial center and endowed with an outer action of G. This outer action uniquely determines an extension Ĝ of G by P. With suitable hypotheses, the scheme H1nab(G,P) parametrizes the P conjugacy classes of sections of Ĝ →G. © 2013 University Press, Inc.