  The present paper is dedicated to the study of the global existence for the
inviscid two-dimensional Boussinesq system. We focus on finite energy data with
bounded vorticity and we find out that, under quite a natural additional
assumption on the initial temperature, there exists a global unique solution.
None smallness conditions are imposed on the data. The global existence issues
for infinite energy initial velocity, and for the B\'enard system are also
discussed.
