In this thesis, we study one of the most important issues occurring in computer games, which can also be found in many other domains, such as virtual reality applications and robotics. We begin by reviewing simple tests. Due to the observed limitations of simple tests, the method of separating axes is introduced, which results in the possibility of collision testing on all convex shapes. We explain the application of the method, and list its strengths. Furthermore, we present the appropriate collision detection algorithms based on the separating axis theorem, and we examine the associated problem of contact sets determination. We extend the algorithms to 3D objects and to cases where objects are moving at constant linear velocity. Moreover, we analyse the problem of tunneling and its potential solutions. The fifth chapter deals with the problem of collision testing on non-convex objects and convex objects of extremely high complexity, which require the construction of a special structure, the so-called bounding boxes. In addition, we discuss several possible optimizations in order to improve the precision and speed of the algorithms. Finally, we demonstrate the effectiveness of the algorithms by creating a program in C++, using the Box2D library for the game physics and Sfml for the game graphics.