In this thesis, a weak coupling formalism is developed to study superconductivity in spin-orbit coupled, multi-orbital systems. This formalism is then applied to Sr2RuO4, one of the few candidates for odd-parity superconductivity. We show that spin-orbit coupling and multi-band effects are crucial to understand the physics of this material. Depending on the interaction parameters, the order parameter can either be chiral or helical. In both cases, the gap is highly anisotropic, and has accidental deep minima along certain directions, in accordance with experiments. Focusing then on the chiral case, we show that the total Chern number is -7 instead of the usually assumed +1. This leads to drastically different predictions for the thermal and charge Hall conductances. In particular, we show that the absence of measurable charge edge currents is not incompatible with a chiral state. Finally, we study the evolution of superconductivity in Sr2RuO4 under ?100? uniaxial strain. We find a good agreement with experiments for our prediction of Tc as a function of strain. Furthermore, we find that (1) the absence of a measurable cusp of Tc at zero strain is not incompatible with a chiral state and that (2) there could be a transition to an even-parity state at larger strain close to a Van Hove singularity. We propose Hc,2/Tc2 c as a measurable quantity to identify this transition.