  The wealth of incoming and future cosmological observations will allow us to
map out the structure and evolution of the observable universe to an
unprecedented level of precision. Among these observations is the weak
gravitational lensing of galaxies, e.g., cosmic shear that measures the minute
distortions of background galaxy images by intervening cosmic structure. Weak
lensing and cosmic shear promise to be a powerful probe of astrophysics and
cosmology, constraining models of dark energy, measuring the evolution of
structure in the universe, and testing theories of gravity on cosmic scales.
However, the intrinsic alignment of galaxies -- their shape and orientation
before being lensed -- may pose a great challenge to the use of weak
gravitational lensing as an accurate cosmological probe, and has been
identified as one of the primary physical systematic biases in cosmic shear
studies. Correlations between this intrinsic alignment and the lensing signal
can persist even for large physical separations, and isolating the effect of
intrinsic alignment from weak lensing is not trivial. A great deal of work in
the last two decades has been devoted to understanding and characterizing this
intrinsic alignment, which is also a direct and complementary probe of
structure formation and evolution in its own right. In this review, we report
in a systematic way the state of our understanding of the intrinsic alignment
of galaxies, with a particular emphasis on its large-scale impact on weak
lensing measurements and methods for its isolation or mitigation. (Abridged)
