  In this review I give an overview on the conceptual issues involved in the
question how to interpret so-called `direct top quark mass measurements', which
are based on the kinematic reconstruction of top quark decay products at the
Large Hadron Collider (LHC). These measurements quote the top mass parameter
$m_t^{\rm MC}$ of Monte-Carlo event generators with current uncertainties of
around $0.5$ GeV. At present time the problem of finding a rigorous relation
between $m_t^{\rm MC}$ and top mass renormalization schemes defined in field
theory is unresolved and touches perturbative as well as nonperturbative
aspects and the limitations of state-of-the-art Monte-Carlo event generators. I
review the status of LHC top mass measurements, illustrate how conceptual
limitations enter and explain a controversy that has permeated the community in
the context of the interpretation problem. Recent advances in acquiring first
principle insights are summarized, and it is outlined what else has to be
understood to fully resolve the issue. For the time being, I give a
recommendation how to deal with the interpretation problem when making top mass
dependent theoretical predictions.
