  We review our theoretical results about the sound propagation in
two-dimensional (2D) systems of ultracold fermionic and bosonic atoms. In the
superfluid phase, characterized by the spontaneous symmetry breaking of the
$U(1)$ symmetry, there is the coexistence of first and second sound. In the
case of weakly-interacting repulsive bosons, we model the recent measurements
of the sound velocities of 39K atoms in 2D obtained in the weakly-interacting
regime and around the Berezinskii-Kosterlitz-Thouless (BKT)
superfluid-to-normal transition temperature. In particular, we perform a quite
accurate computation of the superfluid density and show that it is reasonably
consistent with the experiment. For superfluid attractive fermions, we
calculate the first and second sound velocities across the whole BCS-BEC
crossover. In the low-temperature regime we reproduce the recent measurements
of first-sound speed with 6Li atoms. We also predict that only in the
finite-temperature BEC regime there is mixing between sound modes.
