In this thesis, we study the unconventional superconductivity in Sr2RuO4 (SRO) and related systems. The superconducting state in SRO remains a puzzle after more than 28 years of study. Early experiments had pointed toward a topological non-trivial time-reversal symmetry breaking (TRSB) chiral p-wave order. This pairing candidate has attracted a large amount of attention, partly in relation to the possibility of topological quantum computation, and has stimulated studies on higher chirality superconducting systems. In the first part of this thesis, we study the spontaneous edge current in chiral d- and f-wave superconductors. We show that these currents, which vanish in the continuum limit at zero temperature, are generally non-vanishing but tiny, compared to the simplest chiral p-wave case. In the presence of strong surface roughness, the direction of the edge current in the chiral d-wave case can be reversed, compared with that of a specular ideal surface with specular scattering. However, it is shown that this current reversal is non-universal beyond the continuum limit. The chiral p-wave scenario in SRO is overturned by recent Knight shift measurements, highlighting the importance of exploring different pairing symmetries for SRO. Recently, $d_{x^2-y^2} \pm ig_{(x^2-y^2)xy}$, $s' \pm id_{xy}$ and mixed helical p-wave pairings have been proposed as order parameter candidates. However, the stability of these states, especially of the $d_{x^2-y^2} \pm ig$ pairing, remains unclear. In the second part of the thesis, we study the leading superconducting instabilities in SRO in the presence of sizable atomic spin-orbit coupling (SOC), non-local SOC, and non-local interactions. We find that it is difficult to stabilize chiral p-wave pairing in SRO models; this is because, among the triplet p-wave states, the atomic SOC favors helical states over the chiral state. The presence of both d- and g-wave pairings, including a $d_{x^2-y^2} \pm ig$ state, is found when the second nearest neighbor (in-plane) repulsions, together with orbital-anisotropy of the non-local interactions and/or the B2g channel non-local SOC are included. We further analyze the properties, such as nodal structures, in-plane field spin-susceptibility, and spontaneous edge current, of the realized $d_{x^2-y^2} \pm ig$ pairing and find that this state is more compatible with existing experimental measurements than the $s' \pm id_{xy}$ and the mixed helical p-wave proposals. / Dissertation / Doctor of Philosophy (PhD)