Ethylene is an important feedstock for various industrial processes, particularly in the polymer industry. Unfortunately, during naphtha cracking to produce ethylene, there are instances of acetylene presence in the product stream, which poisons the Ziegler–Natta polymerization catalysts. Thus, appropriate process modification, optimization, and in particular, catalyst design are essential to ensure the production of highly pure ethylene that is suitable as a feedstock in polymerization reactions. Accordingly, carefully selected process parameters and the application of various catalyst systems have been optimized for this purpose. This review provides a holistic view of the recent reports on the selective hydrogenation of acetylene. Previously published reviews were limited to Pd catalysts. However, effective new metal and non-metal catalysts have been explored for selective acetylene hydrogenation. Updates on this recent progress and more comprehensive computational studies that are now available for the reaction are described herein. In addition to the favored Pd catalysts, other catalyst systems including mono, bimetallic, trimetallic, and ionic catalysts are presented. The specific role(s) that each process parameter plays to achieve high acetylene conversion and ethylene selectivity is discussed. Attempts have been made to elucidate the possible catalyst deactivation mechanisms involved in the reaction. Extensive reports suggest that acetylene adsorption occurs through an active single-site mechanism rather than via dual active sites. An increase in the reaction temperature affords high acetylene conversion and ethylene selectivity to obtain reactant streams free of ethylene. Conflicting findings to this trend have reported the presence of ethylene in the feed stream. This review will serve as a useful resource of condensed information for researchers in the field of acetylene-selective hydrogenation.