We carried out measurements of domain wall (DW) velocities driven by magnetic-field pulses in symmetric Pt/Co/Pt and asymmetric Pt/Co/AlOx, Pt/Co/GdOx and Pt/Co/Gd trilayers with ultrathin Co layers and perpendicular magnetic anisotropy. The maximum observed velocity is much larger in the asymmetric samples, where the interfacial Dzyaloshinskii-Moriya interaction (DMI) stabilises chiral Néel walls. In quantitative agreement with analytical models, in all samples the maximum observed DW speed scales as , where D is the strength of the DMI and the spontaneous magnetisation. In Pt/Co/Gd, where the anti-parallel coupling between the magnetic moments of Gd and Co leads to a decrease of the total magnetisation, very large DW speeds (up to 700 m/s) are obtained.