Three-dimensional direct numerical simulations of unforced, incompressible, free, spatially evolving round jets are used to investigate the onset of instability at low diametral Reynolds numbers (Re less than or equal to 500). Compact, coherent structures are identified by means of iso-surfaces of vorticity and pressure fields and shown to be synonymous with instability modes. Once the inflow velocity profile is fixed, as the Reynolds number increases from 200 to 500, the most amplified unstable mode switches from the helical mode to the axisymmetric one, as expected from the predictions of the viscous linear stability theory analysis and from experimental observations [J. Fluid Mech. 77, 511 (1976); Prog. Aerosp. Sci. 21, 159 (1984)][ J. Fluid Mech. 48, 547 (1971)]. At the upper limit of the investigated range of Reynolds numbers, the present simulations are consistent with the widely accepted scenario of the space time development of the round jet instability. This scenario is analyzed in detail. The appearance of pairs of axially counter-rotating vortex filaments is found (for the first time, to our knowledge, in unforced, spatial numerical simulations) to characterize the destabilization of initial axisymmetric vortical structures, The spatial evolution of these structures is investigated and their role in vortex rings reconnection is evidenced. For lower Reynolds numbers, a superposition of symmetry-breaking (helical) modes is shown to characterize the instability of the round jet. The Fourier decomposition of the fluctuating flow field allows the extraction of the helical modes and the identification of the flow patterns resulting from their interactions. The attractor is shown to be a limit torus very close to the onset of the instability