The conventional wisdom in cosmology is that that cold-dark-matter haloes form out of initial density peaks and continuously accrete matter from the surroundings becoming more and more massive with time. These concepts form the basis of several semi-analytical tools that are regularly used to interpret data from galaxy and cluster surveys and develop galaxy-formation models (e.g. the halo mass function, merger trees, galaxy biasing). Using a suite of numerical simulations, we present thorough tests of the established scenario and show that it is mandatory to introduce new elements in the theory of the gravitational collapse of cold matter.