The direct linearization method (DLM) for tolerance analysis of 3-D mechanical assemblies is presented. Vector assembly models are used, based on 3-D vector loops which represent the dimensional chains that produce tolerance stackup in an assembly. Tolerance analysis procedures are formulated for both open and closed loop assembly models. The method generalizes assembly variation models to include small kinematic adjustments between mating parts. Open vector loops describe critical assembly features. Closed vector loops describe kinematic constraints for an assembly. They result in a set of algebraic equations which are implicit functions of the resultant assembly dimensions. A general linearization procedure is outlined, by which the variation of assembly parameters may be estimated explicitly by matrix algebra. Solutions to an over-determined system or a system having more equations than unknowns are included. A detailed example is presented to demonstrate the procedures of applying the DLM to a 3-D mechanical assembly.