The talk is devoted to Toeplitz operators acting on the Bergman space over the unit disk. The main concern is to discuss a recently discovered phenomenon, the existence of a wide family of commutative algebras generated by Toeplitz operators with special classes of symbols. As it turned out such classes of generating symbols admit purely geometric descriptions in terms of pencils of geodesics in the hyperbolic geometry of the unit disk. The complete characterization of all possible commutative algebras of Toeplitz operators essentially uses the Berezin quantization procedure. The spectral properties of corresponding Toeplitz operators will be discussed as well.