  Before the 1970s, precision tests for gravity theories were constrained to
the weak gravitational fields of the Solar system. Hence, only the weak-field
slow-motion aspects of relativistic celestial mechanics could be investigated.
Testing gravity beyond the first post-Newtonian contributions was for a long
time out of reach.
  The discovery of the first binary pulsar by Russell Hulse and Joseph Taylor
in the summer of 1974 initiated a completely new field for testing the
relativistic dynamics of gravitationally interacting bodies. For the first time
the back reaction of gravitational wave emission on the binary motion could be
studied. Furthermore, the Hulse-Taylor pulsar provided the first test bed for
the orbital dynamics of strongly self-gravitating bodies.
  To date there are a number of pulsars known, which can be utilized for
precision test of gravity. Depending on their orbital properties and their
companion, these pulsars provide tests for various different aspects of
relativistic dynamics. Besides tests of specific gravity theories, like general
relativity or scalar-tensor gravity, there are pulsars that allow for generic
constraints on potential deviations of gravity from general relativity in the
quasi-stationary strong-field and the radiative regime.
  This article presents a brief overview of this modern field of relativistic
celestial mechanics, reviews some of the highlights of gravity tests with radio
pulsars, and discusses their implications for gravitational physics and
astronomy, including the upcoming gravitational wave astronomy.
