  The Ly-$\alpha$ forest 1D flux power spectrum is a powerful probe of several
cosmological parameters. Assuming a $\Lambda$CDM cosmology including massive
neutrinos, we find that the latest SDSS DR14 BOSS and eBOSS Ly-$\alpha$ forest
data is in very good agreement with current weak lensing constraints on
$(\Omega_m, \sigma_8)$ and has the same small level of tension with Planck. We
did not identify a systematic effect in the data analysis that could explain
this small tension, but we show that it can be reduced in extended cosmological
models where the spectral index is not the same on the very different times and
scales probed by CMB and Ly-$\alpha$ data. A particular case is that of a
$\Lambda$CDM model including a running of the spectral index on top of massive
neutrinos. With combined Ly-$\alpha$ and Planck data, we find a slight
(3$\sigma$) preference for negative running, $\alpha_s= -0.010 \pm 0.004$ (68%
CL). Neutrino mass bounds are found to be robust against different assumptions.
In the $\Lambda$CDM model with running, we find $\sum m_\nu <0.11$ eV at the
95% confidence level for combined Ly-$\alpha$ and Planck (temperature and
polarisation) data, or $\sum m_\nu < 0.09$ eV when adding CMB lensing and BAO
data. We further provide strong and nearly model-independent bounds on the mass
of thermal warm dark matter. For a conservative configuration consisting of
SDSS data restricted to $z<4.5$ combined with XQ-100 \lya data, we find $m_X >
5.3\;\mathrm{keV}$ (95\%CL).
