<p>Active galactic nuclei (AGN) are believed to be powered by accretion of matter onto a supermassive black hole. A fundamental ingredient in shaping our understanding of AGN is their variability across the entire electromagnetic spectrum. Variability studies have the potential to help us understand the geometry of the emitting regions (in various energy bands), their causal relations, and the physics of the accretion processes. This review focusses on the observational properties of AGN variability in the optical/UV/X-ray bands (where most of the AGN luminosity is emitted) and their dependence on the AGN physical parameters (i.e. mass, luminosity, accretion rate). We also discuss possible interpretations in the context of accreting compact systems, and we review the use of variability as a tool to discover AGN and trace their properties across cosmic time, using both ground and space facilities. Finally, we discuss the opportunities and challenges provided by current and next-generation optical/X-ray surveys, to use variability as an effective tool to probe the growth of supermassive black holes in the Universe.</p>