We study the emergence of synchronized burst activity in networks of neurons with spike adaptation. We show that networks of tonically firing adapting excitatory neurons can evolve to a state where the neurons burst in a synchronized manner. The mechanism leading to this burst activity is analyzed in a network of integrate-and-fire neurons with spike adaptation. The dependence of this state on the different network parameters is investigated, and it is shown that this mechanism is robust against inhomogeneities, sparseness of the connectivity, and noise. In networks of two populations, one excitatory and one inhibitory, we show that decreasing the inhibitory feedback can cause the network to switch from a tonically active, asynchronous state to the synchronized bursting state. Finally, we show that the same mechanism also causes synchronized burst activity in networks of more realistic conductance-based model neurons.