  The Weak Gravity Conjecture holds that in a theory of quantum gravity, any
gauge force must mediate interactions stronger than gravity for some particles.
This statement has surprisingly deep and extensive connections to many
different areas of physics and mathematics. Several variations on the basic
conjecture have been proposed, including statements that are much stronger but
are nonetheless satisfied by all known consistent quantum gravity theories. We
review these related conjectures and the evidence for their validity in the
string theory landscape. We also review a variety of arguments for these
conjectures, which tend to fall into two categories: qualitative arguments
which claim the conjecture is plausible based on general principles, and
quantitative arguments for various special cases or analogues of the
conjecture. We also outline the implications of these conjectures for particle
physics, cosmology, general relativity, and mathematics. Finally, we highlight
important directions for future research.
