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Phase transitions are traditionally understood within Landau's framework of spontaneous symmetry breaking. The discovery of topological phases, however, has established a new paradigm in which quantum states are classified by the topology of their electronic wave functions. In this colloquium, I will introduce the basic concepts of topological band theory and discuss how first-principles calculations have guided the discovery of nonmagnetic and magnetic topological insulators. If time permits, I will conclude by presenting our recent work on orbital topology, highlighting a new topological perspective on the orbital Hall effect.