In this paper, we introduce a new type of arithmetic operation that we have called ``fast rotations'' or ``orthonormal \mu-rotations''. These are methods for orthonormal rotation over a set of fixed angles, with a very low cost in implementation, basically a few shift and add operations as opposed to lengthy {\cordic} operations. We also present the underlying theory for the construction of such fast rotation methods. Furthermore, we give examples where fast rotations have been applied successfully in a wide variety of applications. These include the low-cost and robust implementation of FIR filter banks for image coding, the generation of spherical sample rays in 3D graphics, and the computation of the Eigenvalue decomposition (EVD) and singular value decomposition (SVD).