Conformal Field Theory (CFT) is a Quantum Field Theory, invariant under conformal symmetry. Through this thesis we first review the conformal f ield theory in two dimensions and study it symmetry group given by the Virasoro Algebra. As examples of 2-dimentional CFTs, we calculate the partition functions for free bosons and free fermions on the plane and on the Torus and study modular invariance of thier partition functions. By studying the free boson whose target space is compactified on the circle of radius r=1 we observe that its partition function is equal to partition function of two dirac fermions on the torus. Moreover, we investigate on how a non-linear sigma modelcan be used to define a fermionic system with non-abelian symmetry. For this reason we review the two-dimensional bosonization and by introducing the Wess-Zumino- Witten action and its corresponding current algebra, which is known as the Kac-Moody Algebra, we review a n equivalent bosonic definition for a f ermionic system with non-abelian symmetry. Furthermore, in order to study the effect of compactification on the target space of a q uantum field theory we review the Kaluza-Klien reduction approach and finally we study the spectrum of closed bosonic strings under the troidal compactification s and we review the concept of T-Duality.