Problem solving is an integral part of mathematics and is being used with the aim of improving the teaching and mathematical education of students. In order to demonstrate the importance of gaining problem solving skills, the thesis begins by introducing the educational goals of learning and teaching mathematics and the mathematical processes that are an integral part of the Mathematics Curriculum for elementary schools and high schools. In the second chapter we get acquainted with the basic concepts that relate to problem solving teaching, which are the problem, problem situation and problem solving principle. Listed here are the four steps for problem solving described by the famous mathematician George Polya, followed by a list of problem-solving strategies. Some of these strategies have been used in problem solving that is divided here with dependence on the school grade on problems for the students of higher grades of elementary school and problems for high school students. As students in this period of life are in their adolescence, the changes that occur to them are being described, with an emphasis on cognitive development that enables them to successfully use problem solving. For each school level, at the beginning of the chapter, we briefly familiarize ourselves with the math content that students encounter. This is followed by examples of tasks for which possible solutions are given by applying one or more strategies. Finally, everyday life examples are given to the students for research purposes.