We consider the electrostatic interaction between a spherical macroion of charge Qq and an oppositely, highly charged polyelectrolyte of charge − Nq (N,Q > 0). For N ⩽ Q the polyelectrolyte fully collapses on the macroion, while for N > Q only a partial collapse (up to neutralization of the macroion) is expected. We show, however, that for N > Q, the amount of collapsed polyelectrolyte can be bigger than that required to neutralize the macroion, i.e. the macroion can be overcharged. The overcharging increases with the diameter of the macroion (continuously or through multiple first-order transitions), until a total collapse of the polyelectrolyte takes place. The predictions of this model are then compared with Monte Carlo simulations and a possible mechanism is suggested for the overcharging of the octameric histone in the nucleosome.