In almost all analytical and numerical approaches to the study of global properties of non-flat spacetimes, the existence of a singularity at space-like infinity (i^0) becomes fundamental in the characterization of the asymptotics. However, when looking at this problem from the perspective of an Initial Value Problem specifically adapted to take into consideration the geometry of the singularity (pioneered by Friedrich), lots more can be said about its nature, and the request of regular asymptotics translates to specific assumptions on the initial data. Analytically, this requires the investigation of physically relevant motivation for such regulating assumptions, e.g. in relation to the regularity conditions of some energy-momentum densities (inspiered by a conjecture by Horowitz and Tod); numerically, where i^0 is usually excluded from the evolution by either use of specific cut-offs or hyperboloidal slicing, this calls for the need of a unified framework between the different setups in order to implement the same initial data and compare results. In this talk, after an overview of the problem at i^0, some recent developments on both fronts will be addressed.