Online auctions are arguably one of the most important and distinctly new applications of the internet. The predominant player in online auctions, eBay, has over 18.9 milllion users, and it was the host of over $5 billion worth of goods sold in the year 2000. Using methods from approximate dynamic programming and integer programming, we design algorithms for optimally bidding for a single item online auction, and simultaneous or overlapping multiple online auctions. We report computational evidence using data from eBay's web site from 1772 completed auctions for personal digital assistants and from 4208 completed auctions for stamp collections that show that (a) the optimal dynamic strategy outperforms simple but widely used static heuristic rules for a single auction, and (b) a new approach combining the value functions of single auctions found by dynamic programming using an integer programming framework produces high quality solutions fast and reliably. / Singapore-MIT Alliance (SMA)