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Approximation algorithms have developed in response to the impossibility of solving a great variety of important optimization problems. Too frequently, when attempting to get a solution for a problem, one is confronted with the fact that the problem is NP-hard. This, in the words of Garey and Johnson, means "I can't find an efficient algorithm, but neither can all of these famous people." While this is a significant theoretical step, it hardly qualifies as a cheering piece of news.If the optimal solution is unattainable then it is reasonable to sacrifice optimality and settle for a "good" feasible solution that can be computed efficiently. Of course, we would like to sacrifice as little optimality as possible, while gaining as much as possible in efficiency. Trading-off optimality in favor of tractability is the paradigm of approximation algorithms.The main themes of this book revolve around the design of such algorithms and the "closeness" to optimum that is achievable in polynomial time. To evaluate the limits of approximability, it is important to derive lower bounds or inapproximability results. In some cases, approximation algorithms must satisfy additional structural requirements such as being on-line, or working within limited space. This book reviews the design techniques for such algorithms and the developments in this area since its inception about three decades ago.