Superconductivity without time-reversal or inversion symmetries

To ensure a superconducting instability, key symmetries are required. In three dimensions (3D), these are time-reversal (T) and inversion (I) symmetries. In two dimensions, T and I are joined by TM_z and IM_z symmetries where M_z is a mirror reflection through the basal plane. Here we discuss unusual physics that arises when one or more of these key symmetries are broken. In particular, in 3D, we show that spontaneously breaking T symmetry allows for topologically protected nodal Bogoliubov Fermi surfaces and apply this idea to URu₂Si₂ and Sr₂RuO₄. In 2D, we develop an energetic and topological superconductor classification in the unfamiliar situation that both T and I symmetreis are absent. We apply this to Ising superconductors with in-plane magnetic fields and monolayer FeSe coexisting with antiferromagnetic order.