This tutorial introduces the cross-entropy (CE) method, a versatile approach for solving combinatorial and multi-extremal optimization problems and rare event simulation. The CE method is presented as a simple, efficient, and general technique for solving complex optimization problems, including those in combinatorial optimization and machine learning. It is also valuable for rare event simulation, where small probabilities need to be accurately estimated, such as in reliability analysis or performance analysis of telecommunication systems. The CE method was inspired by an adaptive algorithm for estimating probabilities of rare events in complex stochastic networks. It was later realized that a simple cross-entropy modification of this algorithm could be used not only for rare event estimation but also for solving difficult combinatorial optimization problems (COPs). This is achieved by converting a deterministic optimization problem into a related stochastic optimization problem and using rare event simulation techniques. The CE method involves an iterative procedure with two phases: generating a random data sample and updating the parameters of the random mechanism based on the data. The significance of the CE method lies in its ability to define a precise mathematical framework for deriving fast and optimal updating rules based on advanced simulation theory. The CE method can be applied to both deterministic and stochastic COPs. In stochastic COPs, the objective function is random or needs to be estimated via simulation. Examples of such problems include stochastic scheduling, flow control, and data network routing. The CE method is particularly useful for estimating the probability of rare events, which is essential for ensuring the adequate performance of engineering systems. It has been successfully applied to various problems, including the traveling salesman problem, quadratic assignment problem, and max-cut problem. The CE method is a powerful and practical tool for solving NP-hard problems.This tutorial introduces the cross-entropy (CE) method, a versatile approach for solving combinatorial and multi-extremal optimization problems and rare event simulation. The CE method is presented as a simple, efficient, and general technique for solving complex optimization problems, including those in combinatorial optimization and machine learning. It is also valuable for rare event simulation, where small probabilities need to be accurately estimated, such as in reliability analysis or performance analysis of telecommunication systems. The CE method was inspired by an adaptive algorithm for estimating probabilities of rare events in complex stochastic networks. It was later realized that a simple cross-entropy modification of this algorithm could be used not only for rare event estimation but also for solving difficult combinatorial optimization problems (COPs). This is achieved by converting a deterministic optimization problem into a related stochastic optimization problem and using rare event simulation techniques. The CE method involves an iterative procedure with two phases: generating a random data sample and updating the parameters of the random mechanism based on the data. The significance of the CE method lies in its ability to define a precise mathematical framework for deriving fast and optimal updating rules based on advanced simulation theory. The CE method can be applied to both deterministic and stochastic COPs. In stochastic COPs, the objective function is random or needs to be estimated via simulation. Examples of such problems include stochastic scheduling, flow control, and data network routing. The CE method is particularly useful for estimating the probability of rare events, which is essential for ensuring the adequate performance of engineering systems. It has been successfully applied to various problems, including the traveling salesman problem, quadratic assignment problem, and max-cut problem. The CE method is a powerful and practical tool for solving NP-hard problems.