Topological states of matter support quasiparticle excitations with fractional charge and possibly exotic statistics of the non-Abelian type, known as non-Abelian anyons. Most current experimental attempts to reveal such exotic statistics focus on interference involving edge transport. After a brief introduction of topological states (mostly in the context of fractional quantum Hall effect) in general, in this colloquium we will discuss how one can reveal the non-Abelian quasiparticle statistics using bulk probes. We show that bulk thermopower is a promising way to detect their non-Abelian nature, and measure the quantum dimension (a key parameter that quantifies non-Abelian statistics) of these anyons. This method is particularly effective in the Corbino geometry. We also demonstrate a novel cooling effect associated with them. We discuss application of these ideas to the specific candidate system of fractional quantum Hall liquid at filling factor 5/2, and topological insulator-superconductor hybrid systems. Some of the predicted behavior has been observed in recent experiments, which will also be discussed. This body of work has also motivated further theoretical efforts of using thermal probes to study non-Abelian anyons.