A profound consequence of certain classes of topological quantum states of matter is the possibility of topologically protected degeneracies in the ground state spectrum of the system. No local operators can distinguish topologically degenerate states, allowing them to be particularly robust to decoherence. In this talk, I will discuss some new directions in the pursuit of creating, measuring, and manipulating topological degeneracies, which utilize the physics of gapped boundaries. Some specific results I will discuss include (1) a proposal for creating and reading out topological states without localized zero modes in bilayer Laughlin fractional quantum Hall states; and (2) ideas for how to implement topologically robust unitary transformations when the topological degeneracy is due to disconnected gapped boundaries and there are no localized zero modes.