We start from two problems from elementary geometry: description of closed billiard trajectories of length 3 inside a triangle and the Chapple - Euler formula for the distance between centers of inscribed and circumscribed circles of a triangle. Natural generalizations are discussed as a motivation for further investigations. Basic notions of the theory of Riemann surfaces and the theory of integrable systems are presented. Strong interrelations and interactions between these theories are illustrated on two classes of examples: billiards within quadrics and rigid-body motions. The choice of examples is justified by their historical, geometrical, dynamical and methodical significance. Classical, recent and current results are emphasized.