Electronic excitations above the ground state must overcome an energy gap in superconductors with spatially-homogeneous s-wave pairing. In contrast, inhomogeneous superconductors such as those with magnetic impurities or weak links, or heterojunctions containing normal metals or quantum dots, can host subgap electronic excitations that are generically known as Andreev bound states (ABSs). With the advent of topological superconductivity, a new kind of ABS with exotic qualities, known as Majorana bound state (MBS), has been discovered. In this talk, I will focus on hybrid superconductor-semiconductor nanowires as one of the most flexible and promising experimental platforms to study ABSs and MBSs. I'll discuss how the combined effect of spin-orbit coupling and Zeeman field in these wires triggers the transition from ABSs into MBSs. I'll show theoretical progress beyond minimal models in understanding experiments, including the possibility of a type of robust zero mode that may emerge without a band-topological transition, called quasi-MBS or non-topological MBS in the field. Finally, I'll discuss the role of spatial non-locality, a special property of MBS wavefunctions that, together with non-Abelian braiding, is the key to realizing topological quantum computation.

This work has been recently published as a review in: Nature Reviews Physics 2, 575 (2020)