The paper presents the basic elements and results of the latest theory of collision between bodies in rolling. The theory is based on the newly introduced hypothesis of conservation of the sum of the angular momentum of the motions (the sum of kinetic moments) of the bodies in rolling after the collision in relation to the collision and it defines the collision coefficient using the angular rolling velocities immediately after and immediately before the collision. Analytical expressions are derived for the outgoing angular velocities of each body immediately after the collision as a function of the axial moments of inertia of the mass of each body for corresponding instantaneous axis of rolling, the collision coefficient and the angular velocities of each body immediately before the collision. It is shown how the rolling directions of the body are determined immediately after the collision for different types of collisions. It is shown that the basis of research on projects in basic sciences, led by the author of this paper, obtained a large number of scientific results that were published in journals, one master’s thesis (2010), two doctoral dissertations (1996 and 2011), two monographs, one preprint, while one monograph is in preparation for printing. Each of these titles contains the keyword “vibro-impact system”. The paper provides an overview of individual works. The Master of Science thesis and both doctorates are based on the classical theory of bodies colliding in translational motion, rectilinear and curvilinear, smooth or rough, while the latest results relate to the dynamics of vibro-impact body systems in rolling along curvilinear paths in a stationary or rotating vertical plane around a vertical axis at a constant angular velocity. A new methodology for investigating the dynamics of vibro-impact systems with successive collisions between bodies in rolling using the method of phase trajectory portraits in the phase plane has been defined. The list of literature provides the most significant works of the author of this paper who is also a supervisor and a research mentor of undergraduates and doctoral students.