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In general relativity, the dynamics of the collapse of matter under its own weight can be understood in terms of the competition between gravitational attraction and repulsive internal forces. The endstate of such an isolated system is typically either dispersion to flat space or the collapse to a black hole. Critical gravitational collapse occurs when the attractive and repulsive forces are almost in balance. Near criticality the system exhibits self-similarity, scaling and universality. The author, Michael Pürrer, gives an introduction to the numerical solution of characteristic initial value problems in general relativity. The critical collapse of a massless scalar field coupled to Einstein's equations is numerically investigated in spherical symmetry from both global and local points of view using radial compactification. It is found that self-similarity is observable not just locally but also from future null infinity. A surprising correlation between the radiation signal with the period of the first quasinormal mode is pointed out and an argument is made that for astrophysical observers the relevant falloff rate of power-law tails is that of future null infinity.