  Context. Extragalactic surveys are a key tool for better understanding the
evolution of galaxies. Both deep and wide-field surveys serve to provide a
clearer emerging picture of the physical processes that take place in and
around galaxies, and to identify which of these processes are the most
important in shaping the properties of galaxies. Aims. The Lockman
Spectroscopic Redshift Survey using Osiris (Lockman-SpReSO) aims to provide one
of the most complete optical spectroscopic follow-ups of the far-infrared (FIR)
sources detected by the \textit{Herschel} Space Observatory in the Lockman Hole
(LH) field. The optical spectroscopic study of the FIR-selected galaxies
supplies valuable information about the relation between fundamental FIR and
optical parameters, including extinction, star formation rate, and gas
metallicity. In this article, we introduce and provide an in-depth description
of the Lockman-SpReSO project and of its early results. Methods. We selected
FIR sources from \textit{Herschel} observations of the central 24 arcmin
$\times$ 24 arcmin of the LH field with an optical counterpart up to 24.5
$R_{\rm C}$(AB). The sample comprises 956 \textit{Herschel} FIR sources, plus
188 additional interesting objects in the field. These are point X-ray sources,
cataclysmic variable star candidates, high-velocity halo star candidates, radio
sources, very red quasi-stellar objects, and optical counterparts of
sub-millimetre galaxies. The faint component of the catalogue ($R_{\rm
C}(\mathrm{AB})\geq20$) was observed using the OSIRIS instrument on the 10.4 m
Gran Telescopio Canarias in multi-object spectroscopy (MOS) mode. The bright
component was observed using two multi-fibre spectrographs: the AF2-WYFFOS at
the William Herschel Telescope and the HYDRA instrument at the WYIN telescope.
