  Realizing topological superconductivity and Majorana zero modes in the
laboratory is one of the major goals in condensed matter physics. We review the
current status of this rapidly-developing field, focusing on
semiconductor-superconductor proposals for topological superconductivity.
Material science progress and robust signatures of Majorana zero modes in
recent experiments are discussed. After a brief introduction to the subject, we
outline several next-generation experiments probing exotic properties of
Majorana zero modes, including fusion rules and non-Abelian exchange
statistics. Finally, we discuss prospects for implementing Majorana-based
topological quantum computation in these systems.
