The aim of this paper is to present various generalizations of compactness, to study their characteristics and to see how much they differ from compactness. Compact spaces are enriched by characteristics that are a powerful tool in solving various problems, so it is interesting to isolate those characteristics and see how strong each characteristic individually is. The spaces we have observed are paracompact, metacompact, Lindelof, countable compact, BW-compact, sequential compact and pseudocompact spaces. As a result of observing these spaces and their mutual connections, we get the fact that compactness becomes an equivalent characteristic with most of its generalizations on the class of metacompact normal 1-countable spaces, and therefore on the class of metrizable spaces. Furthermore, in order to obtain a characterization of metrizability, we have also dealt with the metrization problem and with the question of how to weaken the sufficient conditions for metrizability expressed in Uryson’s metrizability theorem. The required characterizations presented at the end of this paper, are the Nagata-Smirnov theorem as well as the Smirnov theorem that characterizes metrizable spaces with the help of paracompactness.