Modern neural networks are often augmented with an attention mechanism, which tells the network where to focus within the input. We propose in this paper a new framework for sparse and structured attention, building upon a max operator regularized with a strongly convex function. We show that this operator is differentiable and that its gradient defines a mapping from real values to probabilities, suitable as an attention mechanism. Our framework includes softmax and a slight generalization of the recently-proposed sparsemax as special cases. However, we also show how our framework can incorporate modern structured penalties, resulting in new attention mechanisms that focus on entire segments or groups of an input, encouraging parsimony and interpretability. We derive efficient algorithms to compute the forward and backward passes of these attention mechanisms, enabling their use in a neural network trained with backpropagation. To showcase their potential as a drop-in replacement for existing attention mechanisms, we evaluate them on three large-scale tasks: textual entailment, machine translation, and sentence summarization. Our attention mechanisms improve interpretability without sacrificing performance; notably, on textual entailment and summarization, we outperform the existing attention mechanisms based on softmax and sparsemax.