  We generalize the Kuramoto model by interpreting the $N$ variables on the
unit circle as eigenvalues of a $N$-dimensional unitary matrix $U$, in three
versions: general unitary, symmetric unitary and special orthogonal. The time
evolution is generated by $N^2$ coupled differential equations for the matrix
elements of $U$, and synchronization happens when $U$ evolves into a multiple
of the identity. The Ott-Antonsen ansatz is related to the Poisson kernels that
are so useful in quantum transport, and we prove it in the case of identical
natural frequencies. When the coupling constant is a matrix, we find some
surprising new dynamical behaviors.
