  The discovery of the Higgs boson is both a milestone achievement for the
Standard Model and an exciting probe of new physics beyond the SM. One of the
most important properties of the Higgs is its mass, a number that has proven to
be highly constraining for models of new physics, particularly those related to
the electroweak hierarchy problem. Perhaps the most extensively studied
examples are supersymmetric models, which, while capable of producing a 125 GeV
Higgs boson with SM-like properties, do so in non-generic parts of their
parameter spaces. We review the computation of the Higgs mass in the Minimal
Supersymmetric Standard Model, in particular the large radiative corrections
required to lift $m_h$ to 125 GeV and their calculation via
Feynman-diagrammatic and effective field theory techniques. This review is
intended as an entry point for readers new to the field, and as a summary of
the current status, including the existing analytic calculations and
publicly-available computer codes.
