Topological insulators and semimetals are newly discovered classes of materials characterized by conducting surface states and anomalous electromagnetic responses, both of which are robust to disorder. Much work has gone into trying to classify and discover such topological materials. Until recently, however, very few of the 200,000 stoichiometric compounds in material databases were suspected to harbor topologically protected phenomena. In this talk, I will show how we can reformulate the conventional band theory of solids to highlight the link between the symmetry of localized electronic states and global topological properties. From this I will derive a predictive classification of topological crystalline phases, well suited for both predictions and ab-initio materials searches. Next, I will present several topologically interesting materials that were found using our theory. Finally, I will show how our theory can be used to understand recent discoveries such as higher order and fragile topology.