This thesis collects work on two sets of objects broadly described as charges on 2-dimensional cross-sections of null infinity, the conformal boundary of asymptotically-flat space-times known as I. The first is the Newman-Penrose constants (NPC), a set of five complex constants assigned to each smooth cut of I⁺. These constants maintain the same value regardless of the cut chosen and are therefore supertranslation-invariant. We generalise and numerically compute the NPC for the first time for space-times evolved from generic initial data, and thereby demonstrate their constancy along the generators of I⁺ for physically reasonable space-times. We then probe the response of the NPC to initial wave profiles with varying amplitudes and deviations from axisymmetry. Finally, barriers to an entirely conformally-covariant construction for the NPC force us to reckon with their physicality.

Next, we consider the historical development of charges conjugate to the BMS group of symmetries at I⁺, leading to the Dray-Streubel and Wald-Zoupas prescriptions. Ad hoc procedures in the original presentation of the Dray-Streubel charge are laid out, and we present a formally similar charge which avoids these ambiguities and reduces to the Dray-Streubel charge by a well motivated procedure. The new charge, constructed using twistors on quiescent sections of I⁺, is shown to be conjugate to a preferred Poincaré subgroup of the BMS group and therefore provides a candidate for a relativistic angular momentum which avoids the problem of supertranslation ambiguity. We consider the ability of this quiescent charge to measure the angular momentum which passes through I⁺ during a burst of radiation while avoiding the corresponding shunting of the Poincaré frame by the passing radiation. At the heart of this new procedure is the formalisation of an ad hoc connection between twistors and the BMS group that has been used by a number of authors. This connection may allow for known twistorial methods to be applied to problems on I ⁺.