Let M_n be a random n \times n unitary matrix with distribution given by Haar measure on the unitary group. Using explicit moment calculations, a general criterion is given for linear combinations of traces of powers of M_n to converge to a Gaussian limit as n \rightarrow \infty . By Fourier analysis, this result leads to central limit theorems for the measure on the circle that places a unit mass at each of the eigenvalues of M_n . For example, the integral of this measure against a function with suitably decaying Fourier coefficients converges to a Gaussian limit without any normalisation. Known central limit theorems for the number of eigenvalues in a circular arc and the logarithm of the characteristic polynomial of M_n are also derived from the criterion. Similar results are sketched for Haar distributed orthogonal and symplectic matrices.
