In the design of discrete part shapes, the specification of tolerance constraints can have major consequences for product quality and cost. Traditional methods for tolerance analysis and synthesis are time-consuming and have limited applicability. The thesis of this work is that geometric design systems based on solid modeling technology can be used to automate the solution of these problems. First, a mathematical theory of tolerances is developed. It is shown that a tolerance specification may be expressed as an "in-tolerance" region of a normed vector space over the reals. A representative selection of both dimensional (plus-minus) and geometric tolerance types is investigated. Using this theoretical framework, methods are developed for the solution of tolerance analysis and synthesis problems. Three methods for tolerance analysis are presented: a linear programming method, a Monte Carlo method, and a least squares fitting method. These methods differ as to linearity assumptions and computational costs. The linear programming method supports a worst-case solution basis. The other two methods also support the solution of tolerance analysis problems on a statistical basis. Next, models are developed for worst-case and statistical tolerance synthesis. It is shown that at least in some cases convex programming methods may be applicable. For both the analysis and the synthesis methods, all necessary geometric relationships are automatically derived from the geometric model. To provide a computational framework, a general strategy for constructive variational geometry is developed. This includes a general scheme for the relative positioning of parts and part features. Three relative positioning operators are described. Methods are given for the modeling of size, orientation, and position variations. Feasibility is demonstrated using an experimental geometric modeling system named GEOTOL. The linear programming and Monte Carlo methods are used in solving tolerance analysis problems for several simple assemblies, as well as for a larger bus bar assembly drawn from an actual product. It is shown that three-dimensional tolerancing problems can be solved by these methods.