  The Gaia Data Release 2 contains the 1st release of radial velocities
complementing the kinematic data of a sample of about 7 million relatively
bright, late-type stars. Aims: This paper provides a detailed description of
the Gaia spectroscopic data processing pipeline, and of the approach adopted to
derive the radial velocities presented in DR2. Methods: The pipeline must
perform four main tasks: (i) clean and reduce the spectra observed with the
Radial Velocity Spectrometer (RVS); (ii) calibrate the RVS instrument,
including wavelength, straylight, line-spread function, bias non-uniformity,
and photometric zeropoint; (iii) extract the radial velocities; and (iv) verify
the accuracy and precision of the results. The radial velocity of a star is
obtained through a fit of the RVS spectrum relative to an appropriate synthetic
template spectrum. An additional task of the spectroscopic pipeline was to
provide 1st-order estimates of the stellar atmospheric parameters required to
select such template spectra. We describe the pipeline features and present the
detailed calibration algorithms and software solutions we used to produce the
radial velocities published in DR2. Results: The spectroscopic processing
pipeline produced median radial velocities for Gaia stars with narrow-band
near-IR magnitude Grvs < 12 (i.e. brighter than V~13). Stars identified as
double-lined spectroscopic binaries were removed from the pipeline, while
variable stars, single-lined, and non-detected double-lined spectroscopic
binaries were treated as single stars. The scatter in radial velocity among
different observations of a same star, also published in DR2, provides
information about radial velocity variability. For the hottest (Teff > 7000 K)
and coolest (Teff < 3500 K) stars, the accuracy and precision of the stellar
parameter estimates are not sufficient to allow selection of appropriate
templates. [Abridged]
