Online advertising is a large and rapidly growing business. The major players in the space, namely advertisers, publishers, and ad exchanges, are developing increasingly sophisticated systems, methods and tools to facilitate, manage, optimize and report on the performance of online advertising marketplaces and campaigns. Developing solutions that are both mathematically sound and practical draws on techniques from a variety of disciplines including machine learning, stochastic optimal control, information retrieval, data mining, natural language processing, and econometrics. In this paper, we provide an overview of the online advertising space, and identify, frame, and describe solution approaches to some of the major computational challenges in the space. We describe specific examples from industry applications, including ad inventory auctions, bidding and allocation strategies for ad inventory, inventory targeting, banner and landing page optimization, and performance estimation.