The underlying goal of this Master's thesis is of laying down, in so far as possible, the foundations for later work in Geometric Stochastic Mechanics. The first part is a presentation of symplectic reduction, going through the momentum map and culminating with an explicit construction of a symplectic form on orbits of the coadjoint action of a Lie group. I have made an effort to be as explicit and precise as possible, reviewing many fundamental concepts so that this paper should be readable by anyone who knows the fundamentals of Hamiltonian mechanics as presented, for example, in chapters 5-7 of "Introduction to Mechanics and Symmetry" by Marsden and Ratiu. The second part conveys an introduction to Brownian motion, presenting some of its fundamental properties, defining the Wiener measure and discussing the weak and strong Markov properties.