The aim of the present talk is to discuss HJM cross currency models that can serve as the basis for the simulation of exposure profiles in the xVA context. Such models need to take into account the asymmetries that arise in the different currency denominations in view of the benchmark reform: for example, while in the EUR area Euribor is still the dominant interest rate benchmark, the situation in the US is much more complex due to the introduction of SOFR and alternative forward looking unsecured rates such as the Bloomberg BSBY or the Ameribor 90T. The impact of the Libor transition on the structure of cross currency swap is also an aspect we would like to address. In summary we would like to: - Provide, in a HJM setting, a unified treatment of forward looking and backward looking rates with and without a credit/liquidity component, i.e. consider a HJM setting for a general underlying index in each currency area. - Properly link such general single currency HJM models by means of cross currency processes that capture the cross currency basis. - Analyze cross currency swaps with arbitrary combinations of interest rate indexes and collateral rates in the different currency areas i.e. with and without Libor discontinuation. This is a joint work with Silvia Lavagnini (BI Oslo)