This talk reviews some aspects of the "angular momentum problem" in general relativity as well as of old and new results in the search for covariant and supertranslation-invariant formulas for the flux of angular momentum and other Lorentz charges in asymptotically flat spacetimes. It addresses in particular three problems with all existing solutions: 1) Supertranslation-invariance and covariance imply invariance under spacetime translations; 2) the flux depends on redundant auxiliary degrees of freedom that cannot be set to zero in all Lorentz frames without breaking Lorentz covariance; 3) supertranslation-invariant Lorentz charges do not generate the transformations of the Bondi mass aspect implied by the isometries of the asymptotic metric. The resolution of these problems leads to new, covariant formulas for the fluxes that make explicit the choice of reference frame underpinning seemingly unambiguous results presented in the current literature.