In this paper (for a preliminary version see [Lect. Notes Comput. Sci. 576, 74-85 (1992; Zbl 0789.68050)]) the author studies the applications of universal hashing to the construction of unconditionally secure authentication codes without secrecy. He generalizes the construction given by \textit{M. N. Wegman} and \textit{J. L. Carter} [J. Comput. Syst. Sci. 22, 265-279 (1981; Zbl 0461.68074)], which is useful when the number of authenticators is exponentially small compared to the number of possible source states (plaintext messages), by formally defining some new classes of hash functions. He presents two lower bounds on the size of certain universal classes of hash functions, gives both direct and recursive constructions for universal classes of hash functions and discusses the implications of this theory to the construction of authentication codes. The presented method of universal hashing to construct authentication codes is used by \textit{J. Bierbrauer}, \textit{T. Johansson}, \textit{G. Kobatianskij} and \textit{B. Smeets} [Lect. Notes Comput. Sci. 773, 331-342 (1994)]. In [J. Combinatorial Design 2, 161-166 (1994)] \textit{Tran Van Trung} gives a design-theoretic characterization of those classes of hash functions that meet one of the lower bounds with equality.