At the heart of this SFB 1170 is the study of the interplay of spin-orbit coupling and electron-electron interactions, which could allow for an unambiguous identification of topological superconducting phases. In the first funding period, we investigated signatures of p-wave pairing in non-sinusoidal supercurrent-phase relations, thermal transport and AC Josephson effects in superconducting hybrid structures on topological insulators (TIs) as well as spectroscopic signatures of d + id superconduc- tivity in hexagonal lattices. In the second funding period, we will focus on two classes of systems: (1) hybrid structures on TIs and two-dimensional electron gases with strong spin-orbit coupling and (2) artificially engineered structures which can be tuned into a topological or unconventional supercon- ducting phase. More specifically, in (1) we will investigate recent experiments performed in topological time-reversal symmetric Josephson junctions on HgTe quantum wells. These experiments show clear signatures of the presence of Majorana modes, i.e. even Shapiro steps and fractional Josephson ra- diation. However, these experimental results are in clear contradiction with theory, which predicts that time-reversal symmetric topological Josephson junctions should be coupled to the quasiparticle continuum, leading to a trivial response. Therefore, the central question for the next period will be to find mechanisms which could lead to a decoupling of the Majorana mode from the quasiparticle continuum and understand the role of time-reversal preserving (for finite systems) and time-reversal breaking mechanisms. For the second class of materials (2), the goal is to predict signatures of unconventional phases (for example, in ruthenates, hexagonal materials close to a van Hove sin- gularity and CuxBi2Se3) using combined microscopic (functional renormalization group theory) and macroscopic approaches (theoretical predictions of realistic scanning tunneling microscope signals). Moreover, we will calculate quasiparticle interference patterns from impurities to further confirm the order parameter. While in TIs and Rashba two-dimensional electron gases, unconventional super- conductivity is mainly ruled by spin-orbit coupling, in the second class of materials it is often driven by electron-electron interactions or the lattice symmetry. Therefore, this project addresses the funda- mental physics question which mechanism leads to a more robust formation of topological supercon- ductivity.