  We review the electronic structure of nickelate superconductors with and
without effects of electronic correlations. As a minimal model we identify the
one-band Hubbard model for the Ni 3$d_{x^2-y^2}$ orbital plus a pocket around
the $A$-momentum. The latter however merely acts as a decoupled electron
reservoir. This reservoir makes a careful translation from {nominal} Sr-doping
to the doping of the one-band Hubbard model mandatory. Our dynamical mean-field
theory calculations, in part already supported by experiment, indicate that the
$\Gamma$ pocket, Nd 4$f$ orbitals, oxygen 2$p$ and {the} other Ni 3$d$ orbitals
are not relevant in the superconducting doping regime. The physics is
completely different if topotactic hydrogen is present or the oxygen reduction
is incomplete. Then, a two-band physics hosted by the Ni 3$d_{x^2-y^2}$ and
3$d_{3z^2-r^2}$ orbitals emerges. Based on our minimal modeling we calculated
the superconducting $T_c$ vs. Sr-doping $x$ phase diagram prior to experiment
using the dynamical vertex approximation. For such a notoriously difficult to
determine quantity as $T_c$, the agreement with experiment is astonishingly
good. The prediction that $T_c$ is enhanced with pressure or compressive
strain, has been confirmed experimentally as well. This supports that the
one-band Hubbard model plus an electron reservoir is the appropriate minimal
model.
