A general marching method for surface⧸surface intersection is described for smooth parametric surfaces defined over rectangular and triangular domains. Surface equations are not required explicitly—only evaluated surface positions and tangents. The algorithm is based on an extension of a marching method presented in [Barnhill et al. '87], and also includes ideas from [Houghton et al. '85]. Our new algorithm permits the intersection of triangular surfaces, and the intersection of surfaces that generate tangent and branch points, and tangent tracks. We include a method for approximating step length, and methods for relaxing intersection points onto surface boundaries. These ideas are discussed in this paper and illustrative examples are provided. Comparisons to existing algorithms are also included.