The connection between geometry and mechanics goes back a long way perhaps even as far back as Archimedes. In the 19th century, at the height of the industrial revolution, many of the prominent mathematicians of the day, Chebyshev, Darboux and Sylvester to name just a few, were studying the geometry of mechanisms and machines. Some of the foundations of modern geometry date back to this period, although it is difficult to tell this from a reading of modern texts in geometry. For most of the 20th century, however, this connection was largely ignored by mathematicians and engineers. But the past few years have seen an increasing number of mathematicians looking at geometrical problems in robotics. A few of them have published their work in robotics journals but most have been content to publish their work in maths journals or at specialised conferences. This special issue of Robotica seeks to expose some of this work to a more general robotics audience and begin a mutually beneficial discussion between the mathematicians, computer scientists and engineers. The idea is to present some of the more advanced material in this area to demonstrate to the wider robotics community the power, range and sophistication of these modern ideas. It is hoped that this special issue will show that modern mathematical techniques have a great deal to offer for the practising engineers. Also that many problems in engineering can give rise to interesting and sophisticated mathematics.