We investigate the nonlinear propagation of intense Airy beams forming filaments in transparent media. We demonstrate the existence of stationary nonlinear Airy beams in a planar geometry. These beams preserve the intensity profile and the transverse acceleration of the Airy peak. We show that stationary propagation is sustained by a continuous energy flux to the main Airy lobe from its neighbors. For powers in the main Airy lobe exceeding a certain threshold, this stationary propagation regime becomes unstable. We extend our results to the 2-dimensional case: Airy beams with high powers in the main lobe reshape into a multifilamentary pattern induced by Kerr and multiphoton nonlinearities. The nucleation of new filaments and their interaction, affects the acceleration of the main Airy lobes. Experiments performed in water corroborate the existence of these two regimes.