  The aim of this paper is to give a precise asymptotic description of some
eigenvalue statistics stemming from random matrix theory. More precisely, we
consider random determinants of the GUE, Laguerre, Uniform Gram and Jacobi beta
ensembles and random characteristic polynomials evaluated at 1 for matrices in
the Circular and Circular Jacobi beta ensembles. We use the framework of
mod-Gaussian convergence to provide quantitative estimates of their logarithmic
behavior, as the size of the ensemble grows to infinity. We establish central
limit theorems, Berry-Esseen bounds, moderate deviations and local limit
theorems. Furthermore, we identify the scale at which the validity of the
Gaussian approximation for the tails breaks. With the exception of the Gaussian
ensemble, all the results are obtained for a continuous choice of the Dyson
parameter, that is for general $\beta>0$. The proofs rely on explicit
computations which are possible thanks to closed product formulas of the
Laplace transforms.

Probability