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

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