The dendritic membrane potential was recently measured for the first time in drug-free, naturally behaving rats over several days. These showed that neuronal dendrites generate a lot of sodium spikes, up to ten times as many as the somatic spikes. These key experimental findings are reviewed here, along with a discussion of computational models, and computational consequences of such intense spike traffic in dendrites. We overview the experimental techniques that enabled these measurements as well as a variety of models, ranging from conceptual models to detailed biophysical models. The biophysical models suggest that the intense dendritic spiking activity can arise from the biophysical properties of the dendritic voltage-dependent and synaptic ion channels, and delineate some computational consequences of fast dendritic spike activity. One remarkable aspect is that in the model, with fast dendritic spikes, the efficacy of synaptic strength in terms of driving the somatic activity is much less dependent on the position of the synapse in dendrites. This property suggests that fast dendritic spikes is a way to confer to neurons the possibility to grow complex dendritic trees with little computational loss for the distal most synapses, and thus form very complex networks with high density of connections, such as typically in the human brain. Another important consequence is that dendritically localized spikes can allow simultaneous but different computations on different dendritic branches, thereby greatly increasing the computational capacity and complexity of neuronal networks.