Chikungunya virus (CHIKV) was discovered more than six decades ago, but has remained poorly investigated. However, after a recent outbreak of CHIK fever in both hemispheres and viral adaptation to new species of mosquitoes, it has attracted a lot of attention. The currently available experimental data suggest that molecular mechanisms of CHIKV replication in vertebrate and mosquito cells are similar to those of other New and Old World alphaviruses. However, this virus exhibits a number of unique characteristics that distinguish it from the other, better studied members of the alphavirus genus. This review is an attempt to summarize the data accumulated thus far regarding the molecular mechanisms of alphavirus RNA replication and interaction with host cells. Emphasis was placed on demonstrating the distinct features of CHIKV in utilizing host factors to build replication complexes and modify the intracellular environment for efficient viral replication and inhibition of the innate immune response. The available data suggest that our knowledge about alphavirus replication contains numerous gaps that potentially hamper the development of new therapeutic means against CHIKV and other pathogenic alphaviruses.