  Motivated by the recent discovery of superconductivity in La$_3$Ni$_2$O$_7$
under pressure, we discuss the basic ingredients of a model that captures its
microscopic physics under pressure tuning. We anchor our description in terms
of the spectroscopic evidence of strong correlations in this system. In a
bilayer Hubbard model including the Ni $3d$ $x^2-y^2$ and $z^2$ orbitals, we
show the ground state of the model crosses over from a low-spin $S=1/2$ state
to a high-spin $S=3/2$ state. In the high-spin state, the two $x^2-y^2$ and the
bonding $z^2$ orbitals are all close to half-filling, which promotes a strong
orbital selectivity in a broad crossover regime of the phase diagram pertinent
to the system. Based on these results, we construct an effective multiorbital
$t$-$J$ model to describe the superconductivity of the system, and find the
leading pairing channel to be an intraorbital spin singlet with a competition
between the extended $s$-wave and $d_{x^2-y^2}$ symmetries. Our results
highlight the role of strong multiorbital correlation effects in driving the
superconductivity of La$_3$Ni$_2$O$_7$.
