This tutorial introduces the cross-entropy (CE) method, a versatile approach for combinatorial and multi-extremal optimization, as well as rare event simulation. The CE method is designed to solve complex optimization problems, such as the traveling salesman problem (TSP), quadratic assignment problem (QAP), and max-cut problem, which are typically static and well-known. In contrast, the buffer allocation problem (BAP) is a noisy estimation problem where the objective function is unknown and needs to be estimated through discrete event simulation. The tutorial highlights the CE method's simplicity, efficiency, and generality, making it valuable for both deterministic and stochastic combinatorial optimization problems. It also emphasizes its application in rare event simulation, where small probabilities must be accurately estimated, such as in reliability analysis and performance analysis of telecommunication systems. The CE method involves an iterative process with two phases: generating random data samples and updating parameters based on the data to produce better samples in subsequent iterations. This method provides a precise mathematical framework for deriving fast and optimal updating rules, similar to other randomized methods like simulated annealing, tabu search, and genetic algorithms. The tutorial covers various applications of the CE method, including stochastic node networks (SNN) and stochastic edge networks (SEN), and discusses its success in solving NP-hard problems. It also emphasizes the method's utility in stochastic scheduling, flow control, and routing of data networks, as well as in simulation-based optimization problems like optimal buffer allocation.This tutorial introduces the cross-entropy (CE) method, a versatile approach for combinatorial and multi-extremal optimization, as well as rare event simulation. The CE method is designed to solve complex optimization problems, such as the traveling salesman problem (TSP), quadratic assignment problem (QAP), and max-cut problem, which are typically static and well-known. In contrast, the buffer allocation problem (BAP) is a noisy estimation problem where the objective function is unknown and needs to be estimated through discrete event simulation. The tutorial highlights the CE method's simplicity, efficiency, and generality, making it valuable for both deterministic and stochastic combinatorial optimization problems. It also emphasizes its application in rare event simulation, where small probabilities must be accurately estimated, such as in reliability analysis and performance analysis of telecommunication systems. The CE method involves an iterative process with two phases: generating random data samples and updating parameters based on the data to produce better samples in subsequent iterations. This method provides a precise mathematical framework for deriving fast and optimal updating rules, similar to other randomized methods like simulated annealing, tabu search, and genetic algorithms. The tutorial covers various applications of the CE method, including stochastic node networks (SNN) and stochastic edge networks (SEN), and discusses its success in solving NP-hard problems. It also emphasizes the method's utility in stochastic scheduling, flow control, and routing of data networks, as well as in simulation-based optimization problems like optimal buffer allocation.