  We introduce kicked $p$-spin models describing a family of transverse
Ising-like models for an ensemble of spin-$1/2$ particles with all-to-all
$p$-body interaction terms occurring periodically in time as delta-kicks. This
is the natural generalization of the well-studied quantum kicked top
($p$=2)[Haake, Ku\'s, and Scharf, Z. Phys. B 65, 381 (1987)]. We fully
characterize the classical nonlinear dynamics of these models, including the
transition to global Hamiltonian chaos. The classical analysis allows us to
build a classification for this family of models, distinguishing between $p=2$
and $p>2$, and between models with odd and even $p$'s. Quantum chaos in these
models is characterized in both kinematic and dynamic signatures. For the
latter we show numerically that the growth rate of the out-of-time-order
correlator is dictated by the classical Lyapunov exponent. Finally, we argue
that the classification of these models constructed in the classical system
applies to the quantum system as well.
