This paper introduces multilevel Monte Carlo methods for approximating high-dimensional parameter-dependent integrals. The method is based on the idea of using multiple levels of discretization to reduce computational cost while maintaining accuracy. The paper surveys the multilevel variance reduction technique introduced by the author and presents its extensions and new developments. It discusses the tools needed for the convergence analysis of vector-valued Monte Carlo methods and provides applications to the stochastic solution of integral equations. The paper begins with an introduction to Monte Carlo methods for high-dimensional problems, highlighting the challenge of approximating whole functions, such as integrals depending on a parameter or the solution of an integral equation on a submanifold. It then presents a simple example of parametric integration and describes the standard one-level approach, followed by the multilevel-splitting approach. The multilevel approach is shown to significantly reduce computational cost while maintaining the same level of accuracy. The paper then presents a general result for multilevel Monte Carlo methods, including conditions for the function space and the approximation operators. It discusses the error estimates for the multilevel Monte Carlo method in Banach spaces and shows that the method is optimal in a broad sense. The paper also provides error estimates for the case of $ q = \infty $, showing that the method remains effective even in this case. Finally, the paper discusses the application of these methods to integral equations, showing how they can be used to compute functionals of the solution of an integral equation depending on a parameter. The paper concludes with a discussion of the theoretical and practical implications of the multilevel Monte Carlo method for high-dimensional problems.This paper introduces multilevel Monte Carlo methods for approximating high-dimensional parameter-dependent integrals. The method is based on the idea of using multiple levels of discretization to reduce computational cost while maintaining accuracy. The paper surveys the multilevel variance reduction technique introduced by the author and presents its extensions and new developments. It discusses the tools needed for the convergence analysis of vector-valued Monte Carlo methods and provides applications to the stochastic solution of integral equations. The paper begins with an introduction to Monte Carlo methods for high-dimensional problems, highlighting the challenge of approximating whole functions, such as integrals depending on a parameter or the solution of an integral equation on a submanifold. It then presents a simple example of parametric integration and describes the standard one-level approach, followed by the multilevel-splitting approach. The multilevel approach is shown to significantly reduce computational cost while maintaining the same level of accuracy. The paper then presents a general result for multilevel Monte Carlo methods, including conditions for the function space and the approximation operators. It discusses the error estimates for the multilevel Monte Carlo method in Banach spaces and shows that the method is optimal in a broad sense. The paper also provides error estimates for the case of $ q = \infty $, showing that the method remains effective even in this case. Finally, the paper discusses the application of these methods to integral equations, showing how they can be used to compute functionals of the solution of an integral equation depending on a parameter. The paper concludes with a discussion of the theoretical and practical implications of the multilevel Monte Carlo method for high-dimensional problems.