Topological Superconductivity is a topic of current interest because of its potential for providing a method to reliably store and manipulate quantum information. The most basic topological superconductor has an underlying Ising topological order, in which zero energy Majorana quasiparticle states are associated with topological defects. We will review recent experimental progress towards realizing those states in one and two dimensional superconducting devices. Ising topological order is too simple to allow universal quantum computation, but the richer Fibonacci topological order is in principle sufficient. We will formulate a theory of a Fibonacci phase of a topological superconductor based on a solvable model of interacting Majorana fermions. This theory provides new insight into the nature of the Fibonacci phase, and predicts a closely related "anti-Fibonacci" phase. We show that Majorana fermions can split into a pair of Fibonacci anyons, and propose an interferometer that directly probes Fibonacci non-Abelian statistics.