Importance sampling (IS) is a flexible, theoretically sound, and simple-to-understand methodology for approximation of moments of distributions. It has been widely used in Bayesian inference, rare event estimation, optimization, reinforcement learning, and many other problems. The core IS principle can be applied in very diverse situations (sometimes in an implicit manner) due to its simplicity. The only requirement is the evaluation of the targeted distribution up to a normalizing constant. The basic mechanism of IS consists of (a) simulating samples from simple proposal densities, (b) weighting the samples by accounting for the mismatch between the targeted and the proposal densities, and (c) approximating the moments of interest with the weighted samples. The performance of IS strongly depends on the choice of the proposal distributions. For that reason, the proposals have to be updated and improved so that samples are generated in regions of interest. In this talk, we will first introduce the basics of IS and multiple IS (MIS), motivating the need of using several proposal densities. Then, the focus will be on adaptive IS (AIS) algorithms, describing an encompassing framework of recent methods in the current literature. Finally, we will briefly discuss the inference in dynamic models and the need of a deeper understanding in IS, so that better sequential Monte Carlo (particle filtering) algorithms can be developed.