The Dzyaloshinskii-Moriya interaction (DMI) has been recently recognized to play a crucial role in allowing fast domain wall dynamics driven by spin-orbit torques and the generation of magnetic Skyrmions. Here, we unveil the main features and microscopic mechanisms of DMI in Co/Pt bilayers via first principles calculations. We find that the large DMI of the bilayers has a dominant contribution from the spins of the interfacial Co layer. This DMI between the interfacical Co spins extends very weakly away from the interface and is associated with a spin-orbit coupling in the adjacent atomic layer of Pt. Furthermore, no direct correlation is found between DMI and proximity induced magnetism in Pt. These results clarify the underlying mechanisms of DMI at interfaces between ferromagnetic and heavy metals and should help optimizing material combinations for domain wall and Skyrmion-based devices.