Attractors and their stability

One of the fundamental problems in dynamics is to understand the attractor of a system, i.e. the set where most orbits spent most of the time. As soon as the existence of an attractor is determined, one would like to understand its stability, i.e. if one changes parameters does one still sees a similar attractor? Attractors of the simplest dynamical systems are well understood, and their stability as well. I will survey the known results and discuss the new developments in the understanding of attractors, coexistence of attractors and their stability for more general dynamics which represents better models of physical phenomena.