Global quantization of pseudo-differential operators on compact Lie groups is
introduced relying on the representation theory of the group rather than on
expressions in local coordinates. Operators on the 3-dimensional sphere and on
group SU(2) are analysed in detail. A new class of globally defined symbols is
introduced giving rise to the usual Hormander's classes of operators
$\Psi^m(G)$, $\Psi^m(S^3)$ and $\Psi^m(SU(2))$. Properties of the new class and
symbolic calculus are analysed. Properties of symbols as well as
$L^2$-boundedness and Sobolev $L^2$--boundedness of operators in this global
quantization are established on general compact Lie groups.