Liquids do not mix easily in microfluidic systems, which are being developed into "labs-on-a-chip" that promise revolutionary applications in biotechnology, chemistry and medicine. Recent studies have suggested that microfluidic stirring via chaotic advection can achieve the efficient mixing required in typical uses. For devices based on continuous flow through microchannels, strategies for inducing chaotic mixing by altering device geometries have been proposed. I will describe a general methodology for introducing chaotic mixing in discrete volume (microdroplet) systems, which allow miniaturization of many standard laboratory protocols that are difficult to realize with continuous flow. The mixing properties of the flows in microdroplets are governed by their symmetries, which give rise to invariant surfaces serving as barriers to transport. Complete three-dimensional mixing by chaotic advection requires destruction of all flow invariants. As an illustration of this idea, I will demonstrate that complete mixing can be obtained in a time-dependent flow produced by motion of a microdroplet along a two-dimensional path and describe the experiments that optically manipulate and mix microdroplets.