Exchange statistics, which characterizes how wavefunctions evolve under particle interchange, is an extremely fundamental property of nature that ultimately underlies most quantum condensed matter phenomena. Interchanging ordinary bosons or fermions, for instance, leaves the wavefunction either unchanged or with an extra overall minus sign. The story becomes much richer in a class of superconducting states supporting so-called "Majorana fermions". Interchanging Majorana fermions produces not only a phase, but can actually transform the wavefunction to a fundamentally different quantum state. Apart from revealing something very profound about nature, Majorana fermions also hold promise for implementing decoherence-free quantum information processing. This talk will survey various promising proposals for engineering states supporting Majorana fermions, using materials such as topological insulators, semiconductors, and ordinary superconductors. I will also describe how 1D systems can be used to meaningfully exchange Majorana fermions, in spite of that fact that exchange statistics is usually viewed as ill-defined in 1D.