  We present a complete symmetry classification of the Sachdev-Ye-Kitaev (SYK)
model with $\mathcal{N}=0$, $1$ and $2$ supersymmetry (SUSY) on the basis of
the Altland-Zirnbauer scheme in random matrix theory (RMT). For $\mathcal{N}=0$
and $1$ we consider generic $q$-body interactions in the Hamiltonian and find
RMT classes that were not present in earlier classifications of the same model
with $q=4$. We numerically establish quantitative agreement between the
distributions of the smallest energy levels in the $\mathcal{N}=1$ SYK model
and RMT. Furthermore, we delineate the distinctive structure of the
$\mathcal{N}=2$ SYK model and provide its complete symmetry classification
based on RMT for all eigenspaces of the fermion number operator. We corroborate
our classification by detailed numerical comparisons with RMT and thus
establish the presence of quantum chaotic dynamics in the $\mathcal{N}=2$ SYK
model. We also introduce a new SYK-like model without SUSY that exhibits hybrid
properties of the $\mathcal{N}=1$ and $\mathcal{N}=2$ SYK models and uncover
its rich structure both analytically and numerically.
