The main aim of this chapter is to give a detailed presentation of the bi-Hamiltonian structure of a very classical integrable system, the n-dimensional-free rigid body with a fixed point. More precisely, it will be shown that the Hamiltonian description of this dynamical system can be enhanced to a bi-Hamiltonian one, which, however, does not come from a PN-structure. This presentation will occupy Sect. 3, where, after the description of this dynamical system, we provide a bi-Hamiltonian representation of the equations of motion. Then we introduce two (infinite) sequences of conserved quantities, the so-called Mishchenko and Manakov integrals, which form the Lenard-Magri chain for the bi-Hamiltonian structure previously introduced. The first two sections of this chapter are devoted to more general topics. More precisely, in the first one, we discuss the geometry of the cotangent bundle of a Lie group, with emphasis on the symplectic aspects, while the second one is dedicated to the so-called Euler equations, which are Hamiltonian equations defined on coadjoint orbits.