Quantum lattice systems with local interactions can exhibit phases that cannot be understood within the conventional symmetry-breaking paradigm. Examples include symmetry-protected phases, such as the Haldane phase and certain topological insulators, as well as systems exhibiting so-called intrinsic topological order in two or more spatial dimensions. Still more exotic phenomena are possible in three dimensions. In this talk, I will give a selective overview of these subjects, discussing the general physical picture, representative models, and some approaches to rigorous mathematical understanding. The talk will be expository, with no new results presented.