Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear. Yet they can fit data almost perfectly through variants of gradient descent algorithms and achieve unexpected levels of prediction accuracy without overfitting. These are formidable results that defy predictions of statistical learning and pose conceptual challenges for non-convex optimization. In this paper, we use methods from statistical physics of disordered systems to analytically study the computational fallout of overparameterization in non-convex binary neural network models, trained on data generated from a structurally simpler but "hidden" network. As the number of connection weights increases, we follow the changes of the geometrical structure of different minima of the error loss function and relate them to learning and generalization performance. A first transition happens at the so-called interpolation point, when solutions begin to exist (perfect fitting becomes possible). This transition reflects the properties of typical solutions, which however are in sharp minima and hard to sample. After a gap, a second transition occurs, with the discontinuous appearance of a different kind of "atypical" structures: wide regions of the weight space that are particularly solution-dense and have good generalization properties. The two kinds of solutions coexist, with the typical ones being exponentially more numerous, but empirically we find that efficient algorithms sample the atypical, rare ones. This suggests that the atypical phase transition is the relevant one for learning. The results of numerical tests with realistic networks on observables suggested by the theory are consistent with this scenario.