The physical properties of homogeneous matter at thermal equilibrium are characterized by its equation of state: a relationship between different thermodynamic quantities. The two-dimensionnal Bose gas is particular in this respect because its equation of state is scale invariant in the presence of weak repulsive atomic interactions. Another remarkable feature of the 2D Bose gas is the existence of a phase transition to a superfluid state at low temperature. In this PhD thesis I present a measurement of the equation of state of the homogeneous 2D Bose gas for three thermodynamic quantities: the reduced pressure, the phase-space density and the entropy per particle. I also present a measurement of the interaction energy of a 2D gas trapped in a harmonic potential. This measurement highlighted the existence of a phase preceding the superfluid phase, in which density fluctuations are strongly reduced. This phase is an essential step in the establishment of the superfluid transition. Finally, I describe the observation of signatures of vortices in 2D Bose gases. These vortices are the key ingredient in the microscopic mechanism of the superfluid transition in two dimensions.
