A set of disordered interacting building blocks may form ordered structures by means of a self-assembling process. An external intervention in the system by adding a chemical species or by applying forces leads to different self-assembly scenarios with the appearance of new structures. For instance, the formation of microtubules, gels, virus capsides, cells and living beings among others takes place by self-assembly under nonequilibrium conditions. A general evolution criterion able to account for why nature selects some structures outside equilibrium and not others is lacking. Nevertheless, progress in the understanding of nonequilibrium self-assembly (NESA) mechanisms has been made thanks to the formulation of models that take particular situations into consideration. We review recent efforts devoted to describing self-assembly out of equilibrium and we provide a reference linking several current concepts in order to help in the development of new models and experimental studies. We hope that the knowledge of the intimate mechanisms leading to the formation of structures will make the implementation of re-configurable and bio-inspired materials possible and give a simpler perspective on the understanding of the emergence of life.