  We describe a recent experiment performed with rubidium atoms ($^{87}$Rb),
aiming at studying the coherence properties of a two-dimensional gas of bosonic
particles at low temperature. We have observed in particular a
Berezinskii--Kosterlitz--Thouless (BKT) type crossover in the system, using a
matter wave heterodyning technique. At low temperatures, the gas is
quasi-coherent on the length scale set by the system size. As the temperature
is increased, the loss of long-range coherence coincides with the onset of the
proliferation of free vortices, in agreement with the microscopic BKT theory.
