Mathematics Department, Universidad de Oviedo, 33007 Oviedo, SpainParticle Swarm Optimization is an evolutionary algorithm that has been applied to many differentengineering and technological problems with considerable success. Since its first publication in1995, it has been continually modified trying to improve its convergence properties. Thus, manyvariants have been proposed in the literature. Some of these variants were related to a particularproblem and had little application outside the field where they have been proposed. Others havebeen used for solving different kind of problems and have enjoyed a longer life. These PSO variantshave been used to solve a wide range of optimization and inverse problems: continuous, discrete,dynamical, multioptima, combinatorial, with and without additional constraints. In this paper webriefly review the history of Particle Swarm Optimization, insisting in the importance of the stochasticstability analysis of the particle trajectories in order to achieve convergence.