The abundant production of top pairs at the LHC has provided an opportunity for accurate top mass measurements. The quoted errors of the LHC experimental collaborations have reached values near 500 MeV, an accuracy of about three parts in a thousand. In spite of this, several theoretical issues, mostly related to the impossibility to define the system of the top decay products even at the theoretical level, have clouded the interpretation of these results. This has led to a number of alternative proposals for top mass measurement strategies, but has also left the most precise results in a sort of "limbo", with the hope that some theoretical work may clarify their meaning in the future. In this talk I will discuss these problems from a theoretical perspective, and present ongoing work aimed at clarifying some of these issues.