Coupling is a key method in probability theory that allows for a comparison of random variables. Coupling is a powerful tool that has been applied in a wide variety of different contexts, for instance, to derive probabilistic inequalities, to prove limit theorems and associated rates of convergence, and to obtain sharp approximations. The course first explains what coupling is and what general framework it fits into. After that a number of applications are described, which illustrate the power of coupling and at the same time serve as a guided tour through some key areas of modern probability theory. The course is intended for master students and PhD students. A basic knowledge of probability theory and measure theory is required. Course notes are available, to be distributed at the beginning of the course.