  The nature of the Cooper pairing in the paradigmatic unconventional
superconductor Sr$_2$RuO$_4$ is an outstanding puzzle in condensed matter
physics. Despite the tremendous efforts made in the past twenty-seven years,
neither the pairing symmetry nor the underlying pairing mechanism in this
material has been understood with clear consensus. This is largely due to the
lack of a superconducting order that is capable of interpreting in a coherent
manner the numerous essential experimental observations. At this stage, it may
be desirable to reexamine the existing theoretical descriptions of
superconducting Sr$_2$RuO$_4$. This review focuses on several recent
developments that may provide some clues for future study. We highlight three
separate aspects: 1) any pairing in the $E_u$ symmetry channel, with which the
widely discussed chiral p-wave is associated, shall acquire a 3D structure due
to spin-orbit entanglement; 2) if the reported Kerr effect is a
superconductivity-induced intrinsic bulk response, the superconductivity must
either exhibit a chiral character, or be complex mixtures of certain set of
helical p-wave pairings; 3) when expressed in a multiorbital basis, the Cooper
pairing could acquire numerous exotic forms that are inaccessible in
single-orbital descriptions. The implications of each of these new perspectives
are briefly discussed in connection with selected experimental phenomena.
