This thesis describes two experiments carried out on a two-dimensional (2D) Bose gas cooled down to quantum degeneracy. In the first one we focus on the equilibrium properties of the gas close to the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, while in the second one we study the out-of-equilibrium dynamics of the system. Our experiments are performed using a gas of $\mathrm{<mrow\; data-mjx-texclass="ORD">39</mrow>}$K atoms confined in a 2D box trap, which guarantees the homogeneous gas density and allows tuneability of the interaction strength.

We begin with the studies of the propagation of sound. For temperatures below the BKT critical temperature, we observe first and second sound, as predicted by the hydrodynamic two-fluid model. From the two temperature-dependent speeds of sound and the established thermodynamic properties of the ultracold 2D Bose gas, we extract the superfluid density as a function of temperature. Our results agree with the predictions of the BKT theory, including the universal jump of the superfluid phase-space-density at the transition point.

The second part of the thesis concerns the emergence of the wave-turbulent cascade, as the gas is driven further and further from equilibrium. We monitor how the cascade builds up from large to small length scales starting from the microscopic dynamics of the discrete low-lying quantum states of the system. By probing the gas on all relevant length and time scales we directly observe the emergence of statistical isotropy under anisotropic forcing, and the self-similar spatio-temporal scaling of the momentum spectrum, which are two key theoretical expectations associated with the development of turbulence.