  Linear statistics of eigenvalues in many familiar classes of random matrices
are known to obey gaussian central limit theorems. The proofs of such results
are usually rather difficult, involving hard computations specific to the model
in question. In this article we attempt to formulate a unified technique for
deriving such results via relatively soft arguments. In the process, we
introduce a notion of `second order Poincar\'e inequalities': just as ordinary
Poincar\'e inequalities give variance bounds, second order Poincar\'e
inequalities give central limit theorems. The proof of the main result employs
Stein's method of normal approximation. A number of examples are worked out,
some of which are new. One of the new results is a CLT for the spectrum of
gaussian Toeplitz matrices.
