We compare and contrast the different points of view of rotation in general relativity, put forward by Mach, Thirring and Lense, and Gödel. Our analysis relies on two tools: i) the Sagnac effect which allows us to measure rotations of a coordinate system or induced by the curvature of spacetime, and ii) computer visualizations which bring out the alien features of the Gödel Universe. In order to keep the paper self-contained, we summarize in several appendices crucial ingredients of the mathematical tools used in general relativity. In this way, our lecture notes should be accessible to researchers familiar with the basic elements of tensor calculus and general relativity.