This chapter surveys methods for generating random variables with a given distribution function using random numbers. Both classical and advanced techniques are presented—including the inverse transform, acceptance-rejection, and composition methods—for a variety of basic distributions such as the normal, Poisson, and Gamma distributions, as well as normal vectors. The chapter also discusses practical algorithms for generating uniform distributions on various geometric objects, such as subsets of \(\mathbb {R}^d\) , spheres, ellipses, d-balls, and ellipsoids. In addition, methods for generating copulas are introduced, enabling the construction of dependent random vectors with prescribed marginal distributions. Throughout, step-by-step algorithms, Python code, and illustrative figures and tables are provided to highlight both the theoretical background and practical aspects of simulation. By the end of the chapter, readers will be equipped with tools for simulating a wide range of random variables needed in Monte Carlo methods.