Abstract: The classical three-body problem studies the motion of a system of three point masses under the influence of (Newtonian) gravitational forces. As an alternative to the traditional methods such as Hamiltonian mechanics (i.e. symplectic geometry) we introduce another approach, still using geometric reduction, but in the framework of Riemannian kinematic geometry. The method uses an idea going back to Jacobi's lectures in the 1840's, where he actually reformulated and geometrized the least action principle. Using symmetry reduction we can (in many cases) study the three-body problem as a second order ODE in three variables. As a concrete example we shall also briefly discuss the triple collision motions, which is really the only singularity of the problem. In this talk I will try to explain some important ideas without going into the details, but for illustrative purposes some explicit calculations will also be presented to show what type of results one can obtain.