Abstract: In this paper we give a characterization of compact topological spaces, in special, of compact zero-dimensional spaces (Theorems 1 and 2), in terms of ultrafilter convergence (Definition 1). Motivated by this characterization, we introduce a general method of compactification, which, in the zero-dimensional case, preserves zero-dimensionality. Since those spaces are uniformizable, the method also yields the respective Cauchy completion. In the general case this compactification can be interpreted as "completion" with respect to the ultrafilter convergence. We also prove that the continuous functions with values in compact zero-dimensional spaces possess an extension property w. r. t. the mentioned compactification Ver menos