In the paper we study limit properties of equicontinuous (nearly periodic) positive operators which transform continuous functions into continuous ones. The domain of definition of the functions is a compact Hausdorff space X. Section 1 contains some preliminary information. In Section 2, positive Markov operators are considered. A decomposition of part of the space X into ergodic sub-parts is obtained, which is analogous to the decomposition of Krylov and Bogolyubov. In the next section eigenfunctions of positive operators are studied which correspond to eigenvalues with maximal absolute values. The theory of Perron-Frobenius is generalized to the situation considered. Section 4 is devoted to the investigation of the asymptotic behavior of the powers $T^n $ of Markov transition operators. Finally, in Section 5, we consider the asymptotic behavior of the convolutions $\nu ^n $, $n = 1,2, \cdots $, of a regular measure on a compact topological subgroup. Some results obtained in the previous sections are us...