  We prove a local central limit theorem (LCLT) for the number of points $N(J)$
in a region $J$ in $\mathbb R^d$ specified by a determinantal point process
with an Hermitian kernel. The only assumption is that the variance of $N(J)$
tends to infinity as $|J| \to \infty$. This extends a previous result giving a
weaker central limit theorem (CLT) for these systems. Our result relies on the
fact that the Lee-Yang zeros of the generating function for $\{E(k;J)\}$ ---
the probabilities of there being exactly $k$ points in $J$ --- all lie on the
negative real $z$-axis. In particular, the result applies to the scaled bulk
eigenvalue distribution for the Gaussian Unitary Ensemble (GUE) and that of the
Ginibre ensemble. For the GUE we can also treat the properly scaled edge
eigenvalue distribution. Using identities between gap probabilities, the LCLT
can be extended to bulk eigenvalues of the Gaussian Symplectic Ensemble (GSE).
A LCLT is also established for the probability density function of the $k$-th
largest eigenvalue at the soft edge, and of the spacing between $k$-th neigbors
in the bulk.
