  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 $H_{0}$, 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
$w_0=-0.9\pm0.18$ and $w_a=-0.5\pm1.7$, and when combining also CMB data we
obtain $w_0=-0.98\pm0.11$and $w_a=-0.30\pm0.4$. 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
$\Omega_k=0.003\pm0.003$, including CMB data. Cosmic chronometers data are
important also to constrain neutrino properties by breaking or reducing
degeneracies with other parameters. We find that $N_{eff}=3.17\pm0.15$, thus
excluding the possibility of an extra (sterile) neutrino at more than
$5\sigma$, and put competitive limits on the sum of neutrino masses, $\Sigma
m_{\nu}< 0.27$ eV at 95% confidence level. Finally, we constrain the redshift
evolution of dark energy, and find w(z) consistent with the $\Lambda$CDM model
at the 40% level over the entire redshift range $0<z<2$. [abridged]
