The invention relates to a turbomachine combustion chamber having an inner wall, an outer wall surrounding the inner wall so as to co-operate therewith to define a space forming a combustion area, a transverse wall interconnecting the inner and outer walls, and fuel injection systems. The inner wall has a plurality of inner steps each extending radially towards the outside of the inner wall, the circumferential spacing between two adjacent inner steps defining an inner cavity. The outer wall includes a plurality of outer steps each extending radially towards the inside of the outer wall, the circumferential spacing between two adjacent inner steps defining an outer cavity. At least some of the inner and outer cavities are fed with air from outside the combustion chamber in a common direction that is circumferential, and with fuel in a direction that is radial.