  The unconventional superconductivity in Sr$_2$RuO$_4$ continues to defy a
unified interpretation. In this paper, we focus on some novel aspects of its
superconducting pairing by exploiting the orbital degree of freedom in this
material. The multi-orbital nature, combined with the symmetry of the orbitals
involved, leads to a plethora of exotic Cooper pairings not accessible in
single-orbital systems. Essential physics is illustrated first using a
two-orbital model with $d_{xz}$- and $d_{yz}$-orbitals. We classify the gap
functions according to the underlying lattice symmetries, analyze the effective
theories of a few representative pairings, and make connections to
Sr$_2$RuO$_4$ in the course. In particular, we show how spin-orbit coupling may
entangle spin-triplet and spin-singlet pairings. For completeness, the
classification is generalized to the three-orbital model involving the
$d_{xy}$-orbital as well. The orbital-basis approach distinguishes from the
itinerant-band description for Sr$_2$RuO$_4$, and hence offers an alternative
perspective to investigate its enigmatic superconducting state.
