The relevance of protein stability for biological function and molecular evolution is widely recognized. Protein stability, however, comes in two flavours: thermodynamic stability, which is related to a low amount of unfolded and partially-unfolded states in equilibrium with the native, functional protein; kinetic stability, which is related to a high free-energy barrier "separating" the native state from the non-functional forms (unfolded states, irreversibly-denatured protein). Such barrier may guarantee that the biological function of the protein is maintained, at least during a physiologically relevant time-scale, even if the native state is not thermodynamically stable with respect to non-functional forms. Kinetic stabilization is likely required in many cases, since proteins often work under conditions (harsh extracellular or crowded intracellular environments) in which deleterious alterations (proteolysis, aggregation, undesirable interactions with other macromolecular components) are prone to occur. Also, kinetic stability may provide a mechanism for the evolution of optimal functional properties. Furthermore, enhancement of kinetic stability is essential for many biotechnological applications of proteins. Despite all this, many published studies focus on thermodynamic stability, partly because it can be easily quantified in vitro for small model proteins and, also, because of the availability of computational algorithms to estimate mutation effects on thermodynamic stability. In this review, the opposite bias is purposely adopted: the experimental evidence supporting widespread kinetic stabilization of proteins is summarized, the role of natural selection in determining this feature is discussed, possible molecular mechanisms responsible for kinetic stability are described and the relation between kinetic destabilization and protein misfolding diseases is highlighted.