This paper considers the problem of placing fiducial limits on an unknown parameter of a population, on the basis of a sample drawn from the population. The concept of fiducial limits is generalized to that of finding a fiducial distribution for the parameter. Necessary conditions for application of fiducial theory are considered. Examples are given to illustrate the methods used. Particular attention is given to the meaning which should be associated with a fiducial probability statement. It is shown that, in many cases, fiducial probability has a meaning related to frequency of events. An example is given to illustrate a case where fiducial probability cannot be given such an interpretation. The relationship of the fiducial distribution to a Bayesian posterior distribution is considered. The use of fiducial theory is shown by applying it to solve two important problems in statistical estimation. / M.S.