  We consider the lattice point problem corresponding to a family of elliptic
paraboloids in $\mathbb{R}^d$ with $d\ge3$ and we prove the expected to be
optimal exponent, improving previous results. This is especially noticeable for
$d=3$ because the optimal exponent is conjectural even for the sphere. We also
treat some aspects of the case $d=2$, getting for a simple parabolic region an
$\Omega$-result that is unknown for the classical circle and divisor problems.
