  We prove a central limit theorem for the linear statistics of one-dimensional
log-gases, or $\beta$-ensembles. We use a method based on a change of variables
which allows to treat fairly general situations, including multi-cut and, for
the first time, critical cases, and generalizes the previously known results of
Johansson, Borot-Guionnet and Shcherbina. In the one-cut regular case, our
approach also allows to retrieve a rate of convergence as well as previously
known expansions of the free energy to arbitrary order.
