After having defined the mathematical background of differential forms in the previous chapter, we use that machinery to define canonical transformations, preserving the structure of the Hamiltonian system. We provide definitions of canonical transformations and define their properties. We also give the reader tools to compute the transformations through four types of generating functions. We then prove the conservation of phase volume and show that the transformation given by the phase space flow of a Hamiltonian system is canonical and symplectic. We conclude by proving Liouville’s theorem on the evolution of the density of states in a Hamiltonian system.