I'll begin by explaining what a tiling space is, and how even a quasi-1-dimensional structure can have a rich topology. In fact, the first Cech cohomology gives a tremendous amount of information about a tiling space. We'll then turn to rotation theory. For orientation-preserving self-homeomorphisms of a circle, Poincare defined a rotation NUMBER and showed how the dynamics of iterating the map depend on whether that number is rational or irrational. For tiling spaces, the analogue of the rotation number is a cohomology class, and there's a subtler definition of irrationality. This is joint work with Jose Aliste-Prieto and with UT alum Betseygail Rand.