  The GRAVITY instrument has been commissioned on the VLTI during 2016 and is
now available to the astronomical community. It is the first optical
interferometer capable of observing sources as faint as magnitude 19 in K-band.
This is possible thanks to the fringe tracker which compensates the
differential piston based on measurements of a brighter off-axis astronomical
reference source. The goal of this paper is to consign the main developments
made in the context of the GRAVITY fringe tracker. This could serve as basis
for future fringe tracking systems. The paper therefore covers all aspects of
the fringe tracker, from hardware, to control software and on-sky observations.
Special emphasis is placed on the interaction between the group delay
controller and the phase delay controller. The group delay control loop is a
simple but robust integrator. The phase delay controller is a state-space
control loop based on an auto-regressive representation of the atmospheric and
vibrational perturbations. A Kalman filter provides optimal determination of
the state of the system. The fringe tracker shows good tracking performance on
sources with coherent K magnitudes of 11 on the UTs and 9.5 on the ATs. It can
track fringes with an SNR level of 1.5 per DIT, limited by photon and
background noises. On the ATs, during good seeing conditions, the optical path
delay residuals can be as low as 75 nm root mean square. On the UTs, the
performance is limited to around 250 nm because of structural vibrations.
