This PhD thesis investigates theoretically the proximity-coupling of superconductivity with ferromagnetism and/or spin-orbit coupling (SOC) in hybrid superconductor systems. The results are summarised in three results chapters which assess the proximity effect in hybrid superconductor systems through calculations of the superconducting phase transition (i.e., the critical temperature T_c and critical fields h_c1 and h_c2).

Chapter 3 investigates a ferromagnetic (F) strip on a thin film superconductor (S) with interfacial Rashba SOC in the Ginzburg-Landau formalism. In the presence of SOC, h_c1 has a positive vortex contribution and a negative contribution from the interaction between vortices and SOC. Since the latter is negative, SOC lowers h_c1. When the SOC is strong enough, h_c1 becomes zero and vortices are generated in the absence of a magnetic field.

Chapter 4 considers the phase diagram of a thin film S with SOC in the Usadel formalism. Comparing infinite films with and without SOC, the SOC renormalises the magnetic field, effectively increasing h_c2. In finite sized samples, singlet-to-triplet conversion results in spin magnetisation at the sample edges. This edge effect suppresses the phase transition. Due to the sample size-dependence, the transition can be controlled in shape-anisotropic samples by rotating the applied magnetic field direction.

Finally, chapter 6 explores an s-wave superconductor (S) / chiral p-wave superconductor (P) junction in the Bogoliubov-de Gennes lattice model. In a S/P junction, the singlet Cooper pairs in S cannot mix with the triplet Cooper pairs in P and T_c of both layers remains the same. However, in a S/F/P junction, F converts singlet pairs to triplet pairs, which boosts the P T_c. By rotating the F layer magnetisation, the singlet pairs convert into a different type of triplet pair state that cannot enter P and T_c is unaffected. Hence, in a S/F/P junction, P T_c is magnetisation-orientation-dependent.