A major trend in noncommutative harmonic analysis is to investigate function spaces related to Fourier analysis (and representation theory) of nonabelian groups. The Fourier algebra and the FourierSteiltjes algebra, which are associated with the regular representation and the universal representation of the ambient group respectively, are important examples of such function spaces. These function algebras encode the properties of the group in various ways; for instance the nonexistence of derivations on such algebras indicates their lack of analytic properties. In these lectures, we overview some important properties of these function algebras. For a locally compact group $G$, the FourierStieltjes algebra of $G$, denoted by $B(G)$, is the set of all the matrix coefficient functions of $G$ equipped with pointwise algebra operations. Eymard proved that $B(G)$ can be identified with the dual of the group $C^*$algebra of $G$. Moreover, the FourierStieltjes algebra together with the norm from the above duality turns out to be a Banach algebra. We also study subspaces of $B(G)$, called $A_pi(G)$, generated by all the matrix coefficient functions of $G$ associated with a fixed unitary representation $\pi$. As an important example of such subspaces, we study the Rajchman algebra of $G$, denoted by $B_0(G)$, which is the set of elements of the FourierStieltjes algebra that vanish at infinity.

