This thesis deals with the topic of linear programming, with a special emphasis on the simplex method. The simplex method is one of the most important algorithms for finding the optimal solution for a linear problem. For the algorithm of the simplex method, the feasible region is essential, along whose edges we move in search of an optimal solution. We stop at the vertices of the feasible region, intending to maximize the value of the objective function. We will explain each step of the algorithm, and support it with examples with solutions. We will emphasize the exceptions of the simplex method and ways of using the algorithm in them. The simplex method is a powerful tool for solving optimization problems in various fields. This is a sufficient condition to give special importance to the applications of the simplex method in real life. Namely, this algorithm can be applied to a wide range of problems, from optimization of production and distribution of resources to distribution of costs, planning of investments, and planning of shift schedules.