We will survey recent algorithmic progress on the well-studiedproblem of learning neural networks from labeled random examples. In the first part of the talk, we will describe efficient algorithms for the basic task of learning single-index models in the presence of adversarially corrupted labels. The underlying approaches rely on first-order methods and their analyses draw on ideas from optimization theory. In the second part of the talk, we will present algorithms and computational lower bounds for the problem of learning one-hidden-layer networks. The developed methods leverage ideas from tensor decomposition and algebraic geometry.