Planning and optimizing the motion of one or several robots poses a wide range of problems. What positions should one powerful robot pick to scan a given area with obstacles? How can we coordinate a group of weaker robots to explore an unknown environment? How can we ensure that a swarm of very simple robots with local capabilities can deal with conflicting global requirements? And how can a particle swarm perform complex operations? We will demonstrate how an appropriate spectrum of algorithmic methods in combination with geometry can be used to achieve progress on all of these challenges.