A pro-aflne algebraic group G, over the algebraically closed field k, is an inverse limit of affine algebraic groups over k [5, p. 11271. This paper studies the cohomology, in the category of rational modules, of prounipotent groups when k has characteristic zero (a finite-dimensional rational module V for a pro-affine group G is an abstract G-module such that the corresponding homomorphism G -P GL( v) is a morphism of pro-atfine groups and in general a rational G-module is a direct limit of finitedimensional ones). This theory closely parallels that of the cohomology of pro-p groups [ 12, Chap. I]: the free prounipotent groups turn out to be precisely those of cohomological dimension ones, and the dimension of the first and second cohomology groups give numbers of generators and relations. In addition, one-relator groups turn out to have cohomological dimension two, parallel to the situation for discrete groups [9]. Finally, we apply our theory to the universal pro-affine hull A of a group r to conclude that if r has a free subgroup of finite index, then the prounipotent radical of A is a free prounipotent group. The paper is divided into five sections. The first contains preliminary material on: the pro-variety structure of prounipotent groups, the existence and description of injective rational modules and cohomology, and various technical results used in the rest of the paper. In the second section we define free prounipotent groups and characterize them as groups of cohomological dimension one. This allows us to show that pro-affine subgroups of free prounipotent groups are free prounipotent. In Section 3 we interpret the lowdimensional cohomology of a free prounipotent group with coefficients in the