Metaheuristic optimization algorithms (MOAs) are computational randomized search processes which draw inspiration from physical and biological phenomena, with an application spectrum that extends to numerous fields, ranging from engineering design to economics. MOAs were originally developed for solving unconstrained NP-complete problems, and hence their application to constrained optimization problems (COPs) requires the implementation of specialized techniques that facilitate the treatment of performance and bound constraints. While considerable research efforts have been oriented towards the development and subsequent enhancement of novel constraint handling techniques (CHTs) for MOAs, a systematic review of such techniques has not been conducted hitherto. This work presents a state-of-the-art review on CHTs used with MOAs and proposes eight novel variants based on the feasibility rules and ε-constrained techniques. The distinctive feature of the new variants is that they consider the level and number of constraint violations, besides the objective function value, for selection of individuals within a population. The novel variant performance is evaluated and compared with that of four well-known CHTs from the literature using the metaheuristic pity beetle algorithm, based upon 20 single-objective benchmark COPs. The computational results highlight the accuracy, effectiveness, and versatility of the novel variants, as well as their performance superiority in comparison with existing techniques, stemming from their distinctive formulation. The complete code can be downloaded from GitHub (https://github.com/nikoslagaros/MOAs-and-CHTs).