Animal behaviour depends on learning to associate sensory stimuli with the desired motor command. Understanding how the brain orchestrates the necessary synaptic modifications across different brain areas has remained a longstanding puzzle. Here, we introduce a multi-area neuronal network model in which synaptic plasticity continuously adapts the network towards a global desired output. In this model synaptic learning is driven by a local dendritic prediction error that arises from a failure to predict the top-down input given the bottom-up activities. Such errors occur at apical dendrites of pyramidal neurons where both long-range excitatory feedback and local inhibitory predictions are integrated. When local inhibition fails to match excitatory feedback an error occurs which triggers plasticity at bottom-up synapses at basal dendrites of the same pyramidal neurons. We demonstrate the learning capabilities of the model in a number of tasks and show that it approximates the classical error backpropagation algorithm. Finally, complementing this cortical circuit with a disinhibitory mechanism enables attention-like stimulus denoising and generation. Our framework makes several experimental predictions on the function of dendritic integration and cortical microcircuits, is consistent with recent observations of cross-area learning, and suggests a biological implementation of deep learning.