Majorana bound states (MBS) are zero-energy excitations that can emerge in topological superconductors and feature a number of intriguing properties. They are indeed one of the simplest examples of non-Abelian anyons, quasiparticles that can occur in 2D systems and whose exotic exchange statistics is neither fermionic nor bosonic.

After an introduction focused on the anyonic nature of MBS, I will show how the latter can be exploited to process quantum information via the implementation of topologically protected single and multi-qubit gates. The development of the so-called topological quantum computing is indeed one of the main motivations behind the intense ongoing research on Majorana physics within the condensed matter community. Despite a great theoretical and experimental effort, however, there is no conclusive proof of the existence of MBS yet. In the second part of the talk, I will therefore focus on some of the main challenges involved in the realization and exploitation of MBS. In this respect, I will show how second-order topological superconductors represent a novel and promising platform to create and manipulate MBS.