Magnet-Superconductor Hybrid (MSH) systems have proven to be versatile platforms for the engineering of topological superconductivity and the ensuing Majorana zero modes, an important step towards the realization of topological quantum computing. In particular, the experimental ability to create MSH system with widely varying magnetic structures -- from ferromagnetic and skyrmion-like to antiferromagnetic – has provided an unprecedented opportunity to manipulate and explore topological phases.
In this talk, I will review some recent progress in the theoretical prediction and experimental realization of novel topological superconducting phases – ranging from topological nodal-point superconductivity to higher order topological superconductors -- in MSH systems. Moreover, I will discuss competing and topological phases in the iron-based superconductor FeSe1-xTex and the emergence of dynamically driven out-of-equilibrium topological phases. Finally, I will show how the real-time manipulation of the magnetic structure in MSH systems provides a new path to braiding MZMs, thus revealing their non-Abelian statistics.