The densities of states and the associated characteristic functions of fixed-trace ensembles (FTEs) of N x N random matrices M-N, (tr(M-N(+) M-N) = constant), are calculated exactly at finite N in the case of real-symmetric matrices and of Hermitian matrices. The exact radial density is calculated at finite N for fixed-trace ensembles of complex matrices with no further restrictions on the entries. The density calculated in the Hermitian case coincides with very recent literature results. The exact finite-N density of states of any ensemble of N x N random matrices of a given symmetry, whose probability density depends only on tr(SN+SN), is simply obtained from the density of the FTE of the same symmetry by a one-dimensional integral.