  In an externally applied magnetic field, ultra-pure crystals of the bilayer
compound Sr$_3$Ru$_2$O$_7$ undergo a metamagnetic transition below a critical
temperature, $T^*$, which varies as a function of the angle between the
magnetic field $H$ and the Ru-O planes. Moreover, $T^*$ approaches zero when
$H$ is perpendicular to the planes. This putative "metamagnetic quantum
critical point", however, is preempted by a nematic fluid phase with order one
resistive anisotropy in the {\it ab} plane. In a "realistic" bilayer model with
moderate strength local Coulomb interactions, the existence of a sharp
divergence of the electronic density of states near a van Hove singularity of
the quasi-one-dimensional bands, and the spin-orbit couplings permitted by the
presence of multiple orbitals result in a mean-field phase diagram which
accounts for many of these experimentally observed phenomena. Although the
spin-orbit coupling is not overly strong, it destroys the otherwise near
perfect Fermi surface nesting and hence suppresses spin-density-wave (SDW)
ordering.
