This Technical Report addresses the incompatibilities that exist in the way that the 3GPP and the ITU have addressed QoS classes for IP-based packet flows. This Technical Report has several purposes. It formally documents important work done earlier in the former Committee T1's T1A1 Subcommittee@ specifically the mapping of QoS classes between the ITU-T's Y.1541/Y.1221 and the 3GPP's TS 23-107 (for UMTS). (This work was captured in various T1A1 contributions@ and in external liaisons to the ITU and the 3GPP@ but it was never documented in an ATIS publication@ so it is done so here.) With the formal documentation of the QoS class mapping between the ITU and UMTS presented@ the next purpose of this document is to examine the same question of QoS class mapping between the ITU and LTE classes. It is shown that@ as was the case for UMTS@ a rigorous mapping that fulfills the numerical performance objectives of both domains is neither meaningful nor possible. As the ultimate goal in the industry is fixed/mobile network convergence@ some form of interworking will be necessary@ and it is the purpose of this document to establish the foundation for quantitative methods capable of supporting this critically needed interworking.