Many reactions, including selective hydrogenation of alkynes, take place on solid surfaces. These reactions are vital in many areas of industry including the manufacture of polymers and fine chemicals such as vitamins, fragrances, and drugs. The choice of a catalyst is a trade-off between activity, selectivity and costs. Palladium-based heterogeneous catalysts are traditionally used for these processes as they provide the activation of hydrogen at room temperatures and offers reasonable selectivity, but these catalysts have a number of practical drawbacks. This review discusses recent research work in the selective hydrogenation of alkynes on palladium-based catalysts, emphasises the mechanism and catalytic materials and important applications including alkyne removal from gas-phase alkene precursors for polymer synthesis and liquid phase selective hydrogenation for the synthesis of fine chemicals. Langmuir-Hinshelwood reaction kinetic models, reaction intermediates, formation of carbonaceous layer, the nature of active sites and the effects of reversible and irreversible adsorbates over Pd surface are discussed as well as the factors affecting catalyst activity and selectivity and how these can be optimised in synthetic protocols for these reactions.