Topological insulators (TIs) are insulating crystals in which the electronic wave functions are topologically twisted in a certain sense. In recent years, a bewildering variety of types of TIs have been proposed. In this talk I will focus on two. First, I will introduce 2D quantum anomalous Hall (QAH) insulators, which exhibit a quantum Hall effect without any external electric field. I will discuss our theoretical and computational efforts to identify new, higher-temperature realizations of the QAH state, and briefly discuss the enormous magnetoelectric effects that may become possible if 3D versions can be found. In the second part of the talk I will discuss several types of 3D "axion insulators" whose surfaces are expected to exhibit a half quantum of anomalous Hall conductivity. While attempts to find examples are still in the early stages, such materials could have strange properties, such as unidirectional electron conduction along the edges where facets of different surface orientation meet.