"Explosive percolation" is said to occur in an evolving network when a macroscopic connected component emerges in a number of steps that is much smaller than the system size. Recent predictions based on simulations suggested that certain Achlioptas processes (much-studied local modifications of the classical mean-field growth model of Erdos and Renyi) exhibit this phenomenon, undergoing a phase transition that is discontinuous in the scaling limit. We show that, in fact, all Achlioptas processes have continuous phase transitions, although related models in which the number of nodes sampled may grow with the network size can indeed exhibit explosive percolation.