In Bayesian signal processing, all the information about the unknowns of interest is contained in their posterior distributions. The unknowns can be parameters of a model, or a model and its parameters. In many important problems, these distributions are impossible to obtain in analytical form. An alternative is to generate their approximations by Monte Carlo-based methods like Markov chain Monte Carlo (MCMC) sampling, adaptive importance sampling (AIS) or particle filtering (PF). While MCMC sampling and PF have received considerable attention in the literature and are reasonably well understood, the AIS methodology remains relatively unexplored. This article reviews the basics of AIS as well as provides a comprehensive survey of the state-of-the-art of the topic. Some of its most relevant implementations are revisited and compared through computer simulation examples.