In their recent papers, A.V. Arutyunov and A.V. Greshnov introduced (q1, q2)-quasimetric spaces and studied their properties: investigated covering mappings between (q1, q2)-quasimetric spaces, established sufficient conditions for the existence of a coincidence point for two mappings acting between (q1, q2)-quasimetric spaces such that one is a covering mapping and the other is Lipschitz continuous, proved Banach’s fixed point theorem, obtained generalizations for multivalued mappings. The class of (q1, q2)-quasimetric spaces is sufficiently wide; it includes quasimetric spaces, b-metric spaces, Carnot-Carath´eodory spaces with Box-quasimetics, L_p-spaces with p ∈ (0, 1), etc. The development of the theory of coincidence points of mappings on (q1, q2)-quasimetric spaces initiated interest in the study of more general f-quasimetric spaces and in generalizing Banach’s fixed point theorem to such spaces. The present paper is a review of these results.