Conformal field theory (CFT) is a quantum field theory with conformal symmetry. In this thesis, we start with introducing CFT. Then we focus on two-dimensional CFT. We explain central charge and we show that the algebra of this theory is Virasoro algebra. We use the structure of this theory to study some free theories. Next we check this theory on a torus, and we calculate partition function for fermionic and bosonic state by using modular transformation. The partition function should be modular invariant under modular transformations. After that we study Affine Kac-Moody algebras. We explain currents and Sugawara construction in this algebra. At the end of this thesis we focus on representation of algebra of currents for . For doing that we use free CFTs and we calculate Sugawara construction and central charge.
