Some of the main experimental observations related to the occurrence of exchange bias in magnetic systems are reviewed, focusing the attention on the peculiar phenomenology associated to nanoparticles with core/shell structure as compared to thin film bilayers. The main open questions posed by the experimental observations are presented and contrasted to existing theories and models for exchange bias formulated up to date. We also present results of simulations based on a simple model of a core/shell nanoparticle in which the values of microscopic parameters such as anisotropy and exchange constants can be tuned in the core, shell and at the interfacial regions, offering new insight on the microscopic origin of the experimental phenomenology. A detailed study of the magnetic order of the interfacial spins shows compelling evidence that most of the experimentally observed effects can be qualitatively accounted within the context of this model and allows also to quantify the magnitude of the loop shifts in striking agreement with the macroscopic observed values.