The object of this paper is to review and illustrate the concept of statistical tolerancing and the methods which are applicable to assigning component tolerances. The approach is primarily theoretical, but the theory is that which is useful in practice. Tolerancing and statistical tolerancing are discussed generally, but the in-depth discussion is confined to two facets: 1) the problem of ascertaining the distribution of the response of a mechanism for given component tolerance distributions and 2) the problem due to the shifting and drifting of component tolerance distributions.Four methods are reviewed for the first: a) stack tolerancing or linear propagation of errors, b) nonlinear propagation of errors, c) numerical integration or quadrature, and d) Monte Carlo. The types of problems for which each approach is useful, and drawbacks and advantages are given. Each is illustrated by applying it to the sameπ-section attenuator. The details of this application not covered herein are available in the literature. All in all, the theory and application of this facet of statistical tolerancing is in good shape, although there are open problems. Unfortunately, the same cannot be said for the second facet discussed. The fundamental difficulty is that this problem has not yet been well posed in a theoretical manner applicable in practice. The problem encountered may be the principal reason why statistical tolerancing has not flourished in manufacturing. It also is the part of tolerancing which interfaces most strongly with quality control. The several attacks known to the author are reviewed.Because of its length this paper will be published in three parts in successive issues. Part I lays the groundwork. Part II reviews the four methods mentioned above, and Part III reviews the other facets mentioned and concludes the paper.