Grain boundaries (GBs) in single layer graphene are extended lattice defects that form at the interface between graphene grains. These defects are common in graphene grown by chemical vapor deposition. In this thesis, I study periodic models of twin grain boundaries, which form when adjacent domains are rotated by an equal and opposite angle theta with respect to the grain boundary line. Through band structure and wavefunction analysis of tight-binding Hamiltonians, I find that, although many twin grain boundaries support localized electronic states, only certain theta=0 grain boundaries host localized modes at zero energy. These zero modes, which exist in specific regions of the GB-projected Brillouin zone, are protected by a chiral symmetry of a limited subspace of the Hamiltonian. Topological band theoretic calculations uncover a geometric phase that predicts which Brillouin zone regions support protected modes. The electronic states localized at grain interfaces do not generally carry current, but in a quantizing magnetic field current can flow along grain boundaries. I identify the mechanism that allows grain boundaries to host transport channels and discuss how it is relevant to extended defects in graphene more generally. The grain boundary transport states that develop in the quantum Hall regime exist in bounded regions in energy, suggesting they can be turned "on" and "off" by tuning a gate voltage. When the transport states are "on," quantum Hall edge modes can be deflected into the grain boundary. Accordingly, I propose two kinds of graphene switches that can selectively reroute charge and spin currents. Finally, I study grain boundaries in the Haldane model and in the Kane-Mele model, where bulk graphene is gapped in the absence of an external magnetic field. In the non-trivial phase of each model, I find GB-localized modes that live in the bulk gap but have a finite energy width. Because the non-trivial phase of the Kane-Mele model is the quantum spin Hall (QSH) insulator, I make predictions about the electronic structure of grain boundaries in 1T' transition metal dichalcogenide QSH insulators. A simple model of 1T'-WSe2 GBs mirrors my graphene results and qualitatively agrees with preliminary experimental data.