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James \cite{Jam} introduced uniform covering maps as an analog of coveringmaps in the topological category. Subsequently Berestovskii and Plaut\cite{BP3} introduced a theory of covers for uniform spaces generalizing theirresults for topological groups \cite{BP1}-\cite{BP2}. Their main concepts arediscrete actions and pro-discrete actions, respectively. In case ofpro-discrete actions Berestovskii and Plaut provided an analog of the universalcovering space and their theory works well for the so-called coverable spaces.As will be seen in Section \ref{SECTION-Comparison}, \cite{BP3} generalizesonly regular covering maps in topology and pro-discrete actions may not bepreserved by compositions.  In this paper we redefine the uniform covering maps and we generalizepro-discrete actions using Rips complexes and the chain lifting property. Weexpand the concept of generalized paths of Krasinkiewicz and Minc\cite{KraMin}.

Rips complexes and covers in the uniform category