We report on the discovery of a new member of the magnetar class, SGR J1935+2154, and on its timing and spectral properties measured by an extensive observational campaign carried out between 2014 July and 2015 March with Chandra and XMM–Newton (11 pointings). We discovered the spin period of SGR J1935+2154 through the detection of coherent pulsations at a period of about 3.24 s. The magnetar is slowing down at a rate of $\dot{P} = 1.43(1)\times 10^{-11}$  s s−1 and with a decreasing trend due to a negative $\ddot{P}$ of −3.5(7) × 10−19 s s−2. This implies a surface dipolar magnetic field strength of ∼2.2 × 1014 G, a characteristic age of about 3.6 kyr and a spin-down luminosity Lsd ∼1.7 × 1034 erg s−1. The source spectrum is well modelled by a blackbody with temperature of about 500 eV plus a power-law component with photon index of about 2. The source showed a moderate long-term variability, with a flux decay of about 25 per cent during the first four months since its discovery, and a re-brightening of the same amount during the second four months. The X-ray data were also used to study the source environment. In particular, we discovered a diffuse emission extending on spatial scales from about 1 arcsec up to at least 1 arcmin around SGR J1935+2154 both in Chandra and XMM–Newton data. This component is constant in flux (at least within uncertainties) and its spectrum is well modelled by a power-law spectrum steeper than that of the pulsar. Though a scattering halo origin seems to be more probable we cannot exclude that part, or all, of the diffuse emission is due to a pulsar wind nebula.