  The cosmological constant and its phenomenology remain among the greatest
puzzles in theoretical physics. We review how modifications of Einstein's
general relativity could alleviate the different problems associated with it
that result from the interplay of classical gravity and quantum field theory.
We introduce a modern and concise language to describe the problems associated
with its phenomenology, and inspect no-go theorems and their loopholes to
motivate the approaches discussed here. Constrained gravity approaches exploit
minimal departures from general relativity; massive gravity introduces mass to
the graviton; Horndeski theories lead to the breaking of translational
invariance of the vacuum; and models with extra dimensions change the
symmetries of the vacuum. We also review screening mechanisms that have to be
present in some of these theories if they aim to recover the success of general
relativity on small scales as well. Finally, we summarise the statuses of these
models in their attempt to solve the different cosmological constant problems
while being able to account for current astrophysical and cosmological
observations.
