  Massive gravity has seen a resurgence of interest due to recent progress
which has overcome its traditional problems, yielding an avenue for addressing
important open questions such as the cosmological constant naturalness problem.
The possibility of a massive graviton has been studied on and off for the past
70 years. During this time, curiosities such as the vDVZ discontinuity and the
Boulware-Deser ghost were uncovered. We re-derive these results in a
pedagogical manner, and develop the St\"ukelberg formalism to discuss them from
the modern effective field theory viewpoint. We review recent progress of the
last decade, including the dissolution of the vDVZ discontinuity via the
Vainshtein screening mechanism, the existence of a consistent effective field
theory with a stable hierarchy between the graviton mass and the cutoff, and
the existence of particular interactions which raise the maximal effective
field theory cutoff and remove the ghosts. In addition, we review some
peculiarities of massive gravitons on curved space, novel theories in three
dimensions, and examples of the emergence of a massive graviton from
extra-dimensions and brane worlds.
