  Continued observations of the Double Pulsar, PSR J0737-3039A/B, consisting of
two radio pulsars (A and B) that orbit each other with a period of 2.45hr in a
mildly eccentric (e=0.088) binary system, have led to large improvements in the
measurement of relativistic effects in this system. With a 16-yr data span, the
results enable precision tests of theories of gravity for strongly
self-gravitating bodies and also reveal new relativistic effects that have been
expected but are now observed for the first time. These include effects of
light propagation in strong gravitational fields which are currently not
testable by any other method. We observe retardation and aberrational
light-bending that allow determination of the pulsar's spin direction. In
total, we have detected seven post-Keplerian (PK) parameters, more than for any
other binary pulsar. For some of these effects, the measurement precision is so
high that for the first time we have to take higher-order contributions into
account. These include contributions of A's effective mass loss (due to
spin-down) to the observed orbital period decay, a relativistic deformation of
the orbit, and effects of the equation of state of super-dense matter on the
observed PK parameters via relativistic spin-orbit coupling. We discuss the
implications of our findings, including those for the moment of inertia of
neutron stars. We present the currently most precise test of general
relativity's (GR's) quadrupolar description of gravitational waves, validating
GR's prediction at a level of $1.3 \times 10^{-4}$ (95% conf.). We demonstrate
the utility of the Double Pulsar for tests of alternative theories by focusing
on two specific examples and discuss some implications for studies of the
interstellar medium and models for the formation of the Double Pulsar. Finally,
we provide context to other types of related experiments and prospects for the
future.
