Neurons in cortex show correlations because they are connected and share inputs, but the functional role of these correlations remains elusive. Recently, networks of binary neurons have been successfully used to describe the synchronous activity patterns observed in retina, but these approaches are not applicable to cortex because of scalability problems. Moreover, analytical descriptions of spike correlations generated by realistic networks are lacking, and yet they are crucial to represent faithfully cortical circuits’ behavior. In this talk, I will describe a theory that quantitatively predicts the correlations emerging from networks of spiking neurons with structured common inputs. We show, using the Fokker-Planck formalism and simulations, that the spiking behavior of the network can be exquisitely captured by a current-based model in which the spikes are replaced by the neuronal instantaneous firing rates. Given the parameters of the network and the structure of common inputs, it is possible to predict with high accuracy the correlation patterns of the neurons. The analytical expressions can also be reversed, allowing inferring neuronal connectivity in the network from the observed spiking behavior. The novel techniques developed open new avenues to the study of spatiotemporal correlations in realistic networks of spiking neurons and to infer effective connectivity using multi-electrode recordings. Our results are also crucial to understand under which conditions neuronal networks become dynamically decorrelated, providing novel views to the study of neural coding.