  In this memoir we extend the theory of global pseudo-differential operators
to the setting of arbitrary sub-Riemannian structures on a compact Lie group.
More precisely, given a compact Lie group $G$, and the sub-Laplacian
$\mathcal{L}$ associated to a system of vector fields $X=\{X_1,\cdots,X_k\}$
satisfying the H\"ormander condition, we introduce a (subelliptic)
pseudo-differential calculus associated to $\mathcal{L},$ based on the
matrix-valued quantisation process developed in [138]. This theory will be
developed as follows. First, we will investigate the singular kernels of this
calculus, estimates of $L^p$-$L^p$, $H^1$-$L^1$, $L^\infty$-$BMO$ type and also
the weak (1,1) boundedness of these subelliptic H\"ormander classes. Between
the obtained estimates we prove subelliptic versions of the celebrated sharp
Fefferman $L^p$-theorem and the Calder\'on-Vaillancourt theorem. The obtained
estimates will be used to establish the boundedness of subelliptic operators on
subelliptic Sobolev and Besov spaces. We will investigate the ellipticity, the
construction of parametrices, the heat traces and the regularisation of traces
for the developed subelliptic calculus. A subelliptic global functional
calculus will be established as well as a subelliptic version of Hulanicki
theorem. The approach established in characterising our subelliptic H\"ormander
classes (by proving that the definition of these classes is independent of
certain parameters) will be also applied in order to characterise the global
H\"ormander classes on arbitrary graded Lie groups developed in [90].
