The theory of random processes is used in a wide variety of applications ranging from modeling physical Brownian motion to control theory. Many stochastic problems of interest in engineering and biology involve random rigid-body motions, which is an example of a Lie group. In this talk, a variety of stochastic phenomena that evolve on Lie groups will be discussed. These include the statistical mechanics of DNA and other biopolymers, mobile robot path planning, and manipulator inverse kinematics. Techniques from noncommutative harmonic analysis (i.e., Fourier analysis on Lie groups) are employed to solve Fokker-Planck equations on Lie groups that arise in applications.