In this diploma thesis we exhibit the main results related to the classical Malfatti’s problem (1803.) of cutting out three cylinders with maximal total volume out of a triangular prism. This stereometric riddle can obviously be reduced to the planimetric problem of placing three circles inside a given triangle so that they cover the largest possible area. Malfatti’s conjecture that the optimal arrangement is achieved for three circles such that each one is tangent to the other two and to two sides of the triangle turned out to be wrong. However, this assumption initiated various constructions of such a configuration of circles, as well as calculations of their position using algebraic and geometric methods. The problem of maximizing the area proved to be very hard and the complete solution was accomplished in the late 20th century. The result shows that a greedy algorithm always yields the optimal arrangement of circles, with the corresponding area larger than the one obtained by the Malfatti’s configuration.