We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model.
These are quantum mechanical models involving $N$ Majorana fermions. The
supercharge is given by a polynomial expression in terms of the Majorana
fermions with random coefficients. The Hamiltonian is the square of the
supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken
supersymmetry at large $N$, but non-perturbatively spontaneously broken
supersymmetry in the exact theory. We analyze the model by looking at the large
$N$ equation, and also by performing numerical computations for small values of
$N$. We also compute the large $N$ spectrum of "singlet" operators, where we
find a structure qualitatively similar to the ordinary SYK model. We also
discuss an ${\cal N}=2$ version. In this case, the model preserves
supersymmetry in the exact theory and we can compute a suitably weighted Witten
index to count the number of ground states, which agrees with the large $N$
computation of the entropy. In both cases, we discuss the supersymmetric
generalizations of the Schwarzian action which give the dominant effects at low
energies.