Let G be a tensor category over a field k of characteristic 0. In his paper [-i], w 7, Delinge had established rather general conditions for G to be the category of representations of an algebraic group. He has shown that if any object neC satisfies the condition A " n = 0 for a sufficiently large n, then 0 is equivalent to the category Rep(G) of representations of a proalgebraic group scheme G over k. For tensor categories over fields of positive characteristic this result is no longer true. In the first part of this paper we present a construction of a class of tensor categories over fields of positive characteristic for which an analogue of Deligne's theorem does not hold. To describe these categories, denote by Gz an algebraic Chevalley group over Z and by G the corresponding group over Fp (for a sufficiently large p). Our category C will be a certain "subquotient" of the category R(G) of finite-dimensional representations of G. More precisely, let S be the full subcategory of R(G) whose objects (called small representations) are direct sums of the irreducible representations of G with highest weights inside the fundamental alcove. Let also SL 2 c G be the principal SLz-subgroup in G and (SLz)Im c SL2 be the infinitesimal neighbourhood of the Cartan subgroup H in SL2. Denote by O the full subcategory of R(G) consisting of representations M of G (called quasiprojective representations) such that the restriction of M to (SL2)m) is projective. The category 0 is a quotient of the subcategory S+PcR(G). In the last section of the paper we show that the same approach can be applied to representations of quantum groups. Let U be the quantum group at a root of prime order p of 1 (which is a Hopf algebra over the ring of integers in a cyclotomic field, see [2, 3]). Starting from the category of representations of U, we construct a semisimple braided category ~(G, p) with the finite number of simple objects. On the other hand, in Wess-Zumino-Witten conformal field theory one encounters the category O(G, m) of integrable representations of the correspond-