Remote sounding of the atmosphere has proved to be a fruitful method of obtaining global information about the atmospheres of the earth and other planets. This book treats comprehensively the inverse problem of remote sounding, and discusses a wide range of retrieval methods for extracting atmospheric parameters of interest from the quantities (thermal emission, for example) that can be measured remotely. Inverse theory is treated in depth from an estimation-theory point of view, but practical questions are also emphasized, such as designing observing systems to obtain the maximum quantity of information, efficient numerical implementation of algorithms for processing large quantities of data, error analysis and approaches to the validation of the resulting retrievals. The book is targeted at graduate students as well as scientists.