The free loops space ΛX of a space X has become an important object of study particularly in the case when X is a manifold. The study of free loop spaces is motivated in particular by two main examples. The first is their relation to geometrically distinct periodic geodesics on a manifold, originally studied by Gromoll and Meyer in 1969. More recently the study of string topology and in particular the Chas-Sullivan loop product has been an active area of research. A complete flag manifold is the quotient of a Lie group by its maximal torus and is one of the nicer examples of a homogeneous space. Both the cohomology and Chas-Sullivan product structure are understood for spaces Sn, CPn and most simple Lie groups. Hence studying the topology of the free loops space on homogeneous space is a natural next step. In the thesis we compute the differentials in the integral Leray-Serre spectral sequence associated to the free loops space fibrations in the cases of SU(n+1)/Tn and Sp(n)/Tn. Study in detail the structure of the third page of the spectral sequence in the case of SU(n) and give the module structure of H*(Λ(SU(3)/T2);Z) and H*(Λ(Sp(2)/T2);Z).