In the previous chapters the reader has familiarized himself with probability spaces. The next step is to consider random variables, that means maps from the state space to the real line which are measurable. By measurable it is meant that the pre-image of a Borel set of a map belongs to the \(\sigma \) -algebra given on the state space. As a consequence, we always can determine the probability of those pre-images as the probability measure is defined on the \(\sigma \) -algebra given on the state space. This provides us with the distribution or push forward measure of the random variable. However the chapter starts with introducing random variables as pointwise limits of measurable simple functions and the above description will be given as a characterization. After considering important properties of random variables, generalizations of measurable maps with values in Rˆd and metric spaces are introduced. Finally the difference between discrete and continuous random variables is clarified.