**Symmetry Protected Topological Phases: Prediction, Construction, and Classification**

Topological insulators, with their gapless conducting edges around a gapped insulating bulk, represent a new form of matter. While their topological nature is usually attributed to free electrons filling a topological band structure, it is highly desirable to understand whether similar topological phenomena exist more generally in interacting systems as well. I will show in this talk that indeed with interactions, a large class of such Symmetry Protected Topological (SPT) Phases are possible, even in boson / spin systems where free particles cannot give rise to interesting topological behavior. The quantum information notion of entanglement plays a key role in revealing the underlying structure of the SPT phases and enabling this prediction. In my talk, I will focus on the exotic edge physics characteristic of such SPT phases, the physical models where they can be realized, and the complete classification of spin / boson SPT phases based on the mathematical notion of group cohomology.