  To maximise the potential for new measurements and discoveries at the LHC,
the machine delivers as high as possible collision rates. As a consequence,
multiple proton-proton collisions occur whenever two bunches cross. Interesting
high-energy (hard) collisions are therefore contaminated by several soft,
zero-bias, ones. This effect known as pileup pollutes the final state of the
collision. It complicates the reconstruction of the objects in this final
state, resulting in increased experimental uncertainties. To reduce these
uncertainties, and improve the quality and precision of LHC measurements,
techniques are devised to correct for the effects of pileup. This document
provides a theoretical review of the main methods used during Run I and II of
the LHC to mitigate pileup effects. I start with an in-depth presentation of
the area--median used for the majority of applications, including several
refinements of the original idea, their practical implementation and an
assessment of their efficiency and robustness. I then focus on several
theoretical calculations that can provide both quantitative and qualitative
information on the area--median approach. In the case of boosted jets, a field
that has seen a wide interest recently, a set of methods, known as grooming
techniques has also been used. I describe these techniques, address their
performance and briefly show that they are amenable to a theoretical, analytic,
understanding. The last part of this review focuses on ideas oriented towards
future pileup mitigation techniques. This includes new methods that have
recently been proposed as well as a large series of alternative ideas. The
latter are yet unpublished and have not received the same amount of
investigation than the former but they have the potential to bring new
developments and further improvement over existing techniques in a future where
pileup mitigation will be crucial.
