There have been many interesting recent results in the area of geometrical methods for path planning in robotics. So it seems very timely to attempt a description of mathematical developments surrounding very elementary engineering tasks. Even with such limited scope, there is too much to cover in detail. Inevitably, our knowledge and personal preferences have a lot to do with what is emphasised, included, or left out. Part I is introductory, elementary in tone, and important for understanding the need for geometrical methods in path planning. Part II describes the results on geometrical constructions that imitate well-known constructions from classical approximation theory. Part III reviews a class of methods where classical criteria are extended to curves in Riemannian manifolds, including several recent mathematical results that have not yet found their way into the literature.