  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).
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.
