In probability theory, various types of convergence of random variables are considered. This chapter defines and investigates almost sure convergence, convergence in probability, and convergence in Lp. Sometimes it is desirable to switch from one type of convergence to another, hence it is important to know the implications between the various types. Besides the probabilistic point of view also the analytical formulation of the convergence of random variables is presented. For example, the Ky-Fan metric characterizes convergence in probability. It turns the space of measurable functions defined on a probability space, after taking equivalence classes, into a complete metric space. The concept of uniform integrability plays an important role in Vitali’s convergence theorem which generalizes Lebesgue’s theorem on dominated convergence.