This chapter summarizes recent advances on the analysis of the optimization landscape of neural network training. We first review classical results for linear networks trained with a squared loss and without regularization. Such results show that under certain conditions on the input-output data spurious local minima are guaranteed not to exist, i.e. critical points are either saddle points or global minima. Moreover, the globally optimal weights can be found by factorizing certain matrices obtained from the input-output covariance matrices.We then review recent results for deep networks with parallel structure, positively homogeneous network mapping and regularization, and trained with a convex loss. Such results show that the non-convex objective on theweights can be lower-bounded by a convex objective on the network mapping. Moreover, when the network is sufficiently wide, local minima of the non-convex objective that satisfy a certain condition yield global minima of both the non-convex and convex objectives, and that there is always a non-increasing path to a global minimizer from any initialization.