This thesis aims to develop a sub-elliptic pseudo-differential calculus on any compact Lie group G. We build an operator class Ψ which forms an algebra of operators. We consider a H¨ormander system on G and its associated sub-Laplacian L. The Sobolev spaces that arise naturally from the sub-elliptic operator L are well known, and we check some important properties. Our symbolic calculus is then developed, we define our symbol classes Sm on G and their associated operator classes Ψm, for m ∈ R. A particular example of these symbol classes, Sm(Q₀), is considered and we show that S m(Q0) is contained in any Sm. The core results of this thesis are then proved.