n the thesis, we give an expanded account of works on the Lebesgue-Fourier algebra of a locally compact group during the years ????-????. Throughout this thesis, G will denote a locally compact group with a fixed left Haar measure. Given a function f on G, the left (right) translation of f by x P G is defined by L(L_x f)(y)=f(x^(-?) y) (resp. (R_x f)(y)=f(yx)) for all y?G. In the second chapter, we define the set of all continuous positive definite functions on G, P(G), and its linear span, B(G). The space B(G) can be identified with the dual of the group C^*-algebra C^* (G), the completion of L^? (G) under its largest C^*-norm. With pointwise multiplication and the dual norm, B(G) is a commutative Banach algebra called the Fourier-Stieltjes algebra of G.