Let (X, d) be a quasi-metric space. A Rus–Hicks–Rhoades (RHR) map f : X → X is one satisfying d(fx, f²x) ≤ αd(x, fx) for every x ∈ X, where α ∈ [0, 1). In our previous work [37], we collected various fixed point theorems closely related to RHR maps. In the present article, we collect almost all the known results about RHR maps and their examples. Moreover, we derive new classes of generalized RHR maps and fixed point theorems on them. Consequently, many of the known results in metric fixed point theory are improved and reproved in an easy way.