We present a measurement of the Hubble constant (H-0) and other cosmological parameters from a joint analysis of six gravitationally lensed quasars with measured time delays. All lenses except the first are analysed blindly with respect to the cosmological parameters. In a flat Lambda cold dark matter (Lambda CDM) cosmology, we find H-0 = 73.3(-1.8)(+1.7) km s(-1) Mpc(-1), a 2.4 per cent precision measurement, in agreement with local measurements of H-0 from type Ia supernovae calibrated by the distance ladder, but in 3.1 sigma tension with Planck observations of the cosmic microwave background (CMB). This method is completely independent of both the supernovae and CMB analyses. A combination of time-delay cosmography and the distance ladder results is in 5.3 sigma tension with Planck CMB determinations of H-0 in flat Lambda CDM. We compute Bayes factors to verify that all lenses give statistically consistent results, showing that we are not underestimating our uncertainties and are able to control our systematics. We explore extensions to flat Lambda CDM using constraints from time-delay cosmography alone, as well as combinations with other cosmological probes, including CMB observations from Planck, baryon acoustic oscillations, and type Ia supernovae. Time-delay cosmography improves the precision of the other probes, demonstrating the strong complementarity. Allowing for spatial curvature does not resolve the tension with Planck. Using the distance constraints from time-delay cosmography to anchor the type Ia supernova distance scale, we reduce the sensitivity of our H-0 inference to cosmological model assumptions. For six different cosmological models, our combined inference on H-0 ranges from similar to 73 to 78 km s(-1) Mpc(-1), which is consistent with the local distance ladder constraints.