The commonly adopted classification of the electronic phases of matter as metals, insulators, semimetals, semiconductors and superconductors can be refined by studying the topological properties of the band structure in these phases. This unveils a rich and diverse substructure, and helps to conclude the existence of topological phases---phases whose properties are robust against perturbations, possibly with the requirement that the perturbations preserve certain symmetries. In this dissertation, several topological phases are studied, and predictions are made for some unusual physical phenomena involving these phases. Some of these phenomena may be observable using current experimental techniques. Topological phases often carry unconventional surface states; experimental signatures for the surface states of a strong topological insulator are presented as well.;We begin by systematically discussing the gapped phases, including some topological insulators and topological superconductors, proximate to a three-dimensional Dirac node. Studying topological defects in these phases reveals interesting duality relations between them, which provide potential routes for unconventional or non-Landau-Ginzburg-Wilson continuous phase transitions between them. Topological textures, on the other hand, can lead to very different physics, and we show that a bulk superconductor vortex in a topological insulator---a texture with a non-trivial Hopf index---acts like a fermion. As a corollary, we show that if the vortex string is terminated by a surface, the resulting dangling end traps a Majorana zero mode. The Majorana mode is found to be stable against doping the topological insulator, provided the doping is below a critical value set by Berry phase properties of the Fermi surface of the doped insulator. Importantly, several existing superconductors such as Cu-doped Bi2Se 3 and TlBiTe2 are predicted to host surface Majorana modes. In the absence of superconductivity, the surface of a topological insulator has gapless, spin-momentum locked states. This locking leads to an interesting response to circularly polarized radiation, with the dominant photocurrent being tied to the Berry curvature of the surface bands.;Having studied several theoretical and experimental implications of gapped topological phases, we turn our attention towards a gapless topological phase---the Weyl semimetal. We investigate transport in this phase in the presence of Coulomb interactions or disorder at finite temperature, and compare our results to experimental data on the candidate Weyl semimetal Y2Ir 2O7, finding encouraging agreement.
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