The book explains in very simple terms the need and uses of process capability indices and also identifies the shortcomings of existing and classical approaches when analysing manufacturing process data with single and multiple sources of variability. The concept of a capability index is introduced and the basic theory is extended, with an emphasis on the Bayesian approach. The book also investigates various methods of applying capability indices in various kinds of settings. A large number of indices were studied from a Bayesian point of view. The posterior analysis were done using mainly simulation techniques. Vague information was assumed in the prior distributions of the relevant parameters throught. A vague general non-informative prior is used initially for the joint distribution of the mean and variance. Jeffreys'' priors as well as the popular and mathematically more challenging reference and probability matching priors were also derived for the mean and variance. The analysis should shed some light on the Bayesian approach in Statiscal Quality Control to both practioners and academics.