The promotion of hydrogenation reactions by transitionmetal-based heterogeneous catalysts was established many decades ago but is still quite common in the chemical industry. Because of their importance, these processes have been studied in great detail from both fundamental and practical points of view, and much has been learned about them. However, some key questions remain unanswered, and solutions to specific industrial needs are still pending. In this Perspective, we discuss the state-of-the-art of our understanding of some of the fundamental issues associated with hydrogenation catalysis. From the mechanistic point of view, we use the example of olefin hydrogenation to assess the status of our knowledge on the adsorption of the organic reactants, the role that the strongly adsorbed carbonaceous deposits that form during reaction play in defining the catalytic kinetics, the mechanistic details of the hydrogen dissociative uptake and surface mobility during reaction, and the dynamic changes of the structure of the surface induced by the catalytic conditions. We then introduce the issue of selectivity in connection with the hydrogenation of alkynes, dienes, trienes, and aromatics; unsaturated aldehydes and imines; and cases where hydrogenation competes with other types of reactions such as dehydrogenations, skeletal rearrangements, cyclizations, and hydrogenolysis. Two general approaches to the control of selectivity are discussed: via the tuning of the structure of the catalytic surface, which can now be addressed by using nanoparticles with specific sizes and shapes, and by modifying the electronic properties of the metal, via the addition of a second element. Finally, we make reference to the interest in designing enantioselective hydrogenation processes using heterogeneous metal catalysts, and we briefly summarize the ideas that have developed from surface-science studies toward this goal and the advances made in understanding the most promising approach to date, which involves the addition of molecular chiral modifiers to the reaction mixtures.