We use the latest compilation of observational H(z) measurements obtained with cosmic chronometers in the redshift range 0 < z < 2 to place constraints on cosmological parameters. We consider the sample alone and in combination with other state-of-the art cosmological probes: CMB data from the latest Planck 2015 release, the most recent estimate of the Hubble constant H0, a compilation of recent BAO data, and the latest SNe sample. Since cosmic chronometers are independent of the assumed cosmological model, we are able to provide constraints on the parameters that govern the expansion history of the Universe in a way that can be used to test cosmological models. We show that the H(z) measurements obtained with cosmic chronometer from the BOSS survey provide enough constraining power in combination with CMB data to constrain the time evolution of dark energy, yielding constraints competitive with those obtained using SNe and/or BAO. From late-Universe probes alone we find that w0=−0.9±0.18 and wa=−0.5±1.7, and when combining also CMB data we obtain w0=−0.98±0.11and wa=−0.30±0.4. These new constraints imply that nearly all quintessence models are disfavoured, only phantom models or a pure cosmological constant being allowed. For the curvature we find Ωk=0.003±0.003, including CMB data. Cosmic chronometers data are important also to constrain neutrino properties by breaking or reducing degeneracies with other parameters. We find that Neff=3.17±0.15, thus excluding the possibility of an extra (sterile) neutrino at more than 5σ, and put competitive limits on the sum of neutrino masses, Σmν<0.27 eV at 95% confidence level. Finally, we constrain the redshift evolution of dark energy, and find w(z) consistent with the ΛCDM model at the 40% level over the entire redshift range 0 < z < 2.