A review is presented of recent work related to strange attractors and chaotic motions of dynamical systems. First, simple systems capable of displaying chaotic behavior are discussed. In order of increasing dimensionality of the system, they are one-dimensional noninvertible maps, two-dimensional invertible maps, and autonomous systems of three coupled ordinary differential equations. The concept of fractional dimension of the strange attractor is stressed. Several physical examples well be reviewed, along with the possible relevance to turbulence in systems, such as fluids or plasmas, that are described by partial differential equations.