We introduce a novel approach, requiring only mild assumptions, for the characterization of deep neural networks at initialization. Our approach applies both to fully-connected and convolutional networks and easily incorporates batch normalization and skip-connections. Our key insight is to consider the evolution with depth of statistical moments of signal and noise, thereby characterizing the presence or absence of pathologies in the hypothesis space encoded by the choice of hyperparameters. We establish: (i) for feedforward networks, with and without batch normalization, the multiplicativity of layer composition inevitably leads to ill-behaved moments and pathologies; (ii) for residual networks with batch normalization, on the other hand, skip-connections induce power-law rather than exponential behaviour, leading to well-behaved moments and no pathology.