I will discuss the phenomenon of shear-induced chaos in driven dynamical systems. The unforced system is assumed to be nonchaotic with certain simple structures (such as attracting periodic orbits). Specifics of the defining equations are unimportant. A geometric mechanism for producing chaos - equivalently promoting mixing - is proposed. This mechanism involves the amplification of the effects of the forcing by shearing in the unforced system. Rigorous results establishing the presence of strange attractors will be discussed. Statistical information is deduced by comparing these attractors to countable-state Markov chains. The phenomenon of shear-induced chaos manifests itself in many different guises. Examples presented will include periodically kicked oscillators, slow-fast systems, PDEs undergoing Hopf bifurcations and coupled oscillators.