Pseudo-differential operators on a manifold is an important computational tool in geometry. They are a generalization of differential operators and, in some sense, admit an extension to the Hilbert space of square integrable functions. In this sense, it proves its importance in operator theory. In the seminal work of M. Ruzhansky et al., a very suitable pseudo-differential calculus on compact Lie groups was constructed by using the Peter-Weyl decomposition of L^2 functions. In this seminar, I plan to discuss basic notions of pseudo-differential calculus. I will, then, explain Ruzhansky's framework and present my generalization of it to the deformation quantization of compact quantum groups.