In this article we present Supervised Semantic Indexing which defines a class of nonlinear (quadratic) models that are discriminatively trained to directly map from the word content in a query-document or document-document pair to a ranking score. Like Latent Semantic Indexing (LSI), our models take account of correlations between words (synonymy, polysemy). However, unlike LSI our models are trained from a supervised signal directly on the ranking task of interest, which we argue is the reason for our superior results. As the query and target texts are modeled separately, our approach is easily generalized to different retrieval tasks, such as cross-language retrieval or online advertising placement. Dealing with models on all pairs of words features is computationally challenging. We propose several improvements to our basic model for addressing this issue, including low rank (but diagonal preserving) representations, correlated feature hashing and sparsification. We provide an empirical study of all these methods on retrieval tasks based on Wikipedia documents as well as an Internet advertisement task. We obtain state-of-the-art performance while providing realistically scalable methods.