Gives a complete account of the mathematical structure, statistical analysis, and applications of finite mixture distributions. Direct applications include economics, medicine, remote sensing, sedimentology, and signal detection (pattern recognition). Also describes indirect applications — in outlier models, density estimation, Bayesian and empirical Bayes analysis, and robustness studies. Goes on to cover mathematical concepts such as identifiability and information, and the inferential problems associated with data from a mixture. Approximate sequential methods are developed here, in order to deal with estimation difficulties and engineering applications.