What is the top quark mass?

This review provides an overview of the conceptual issues regarding the interpretation of 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 mMC t of Monte Carlo event generators with current uncertainties of around 0.5 GeV. The problem of finding a rigorous relation between mMC t and top mass renormalization schemes defined in field theory is unresolved to date 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 the picture, and explain a controversy that has permeated the community in the context of the interpretation problem related to mMC t . I then summarize recent advances in acquiring first principles insights and outline what else has to be understood to fully resolve the issue. I conclude with recommendations on how to deal with the interpretation problem for the time being when making top mass-dependent theoretical predictions.