Topological chaos is a type of chaotic behavior that is forced by the motion of obstacles in some domain. I will review two approaches to topological chaos, with applications in particular to stirring and mixing in fluid dynamics. The first approach involves constructing devices where the fluid motion is topologically complex, usually by imposing a specific motion of stirring rods. I will then discuss optimization strategies that can be implemented. The second approach is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits and their topological properties. Many tools and concepts from topological surface dynamics have direct applications: mapping class groups, braids, the Thurston-Nielsen classification theorem, topological entropy, coordinates for equivalence classes of loops, and the Bestvina-Handel algorithm for train tracks.