In this thesis, we review the concepts of conformal field theory. Conformally invariant quantum field theories describe the critical behavior of systems at second order phase transitions. Also Conformal invariants in string theory turns out to give constrains on the allowed space-time dimension and the possible internal degrees of freedom. We study the basic properties of conformal field theories including the discussion of conformal transformation and conformal group in arbitrary dimension and then in two dimension. Also, including the primary fields, radial quantization, the operator product expansion, the operator algebra of quasi-primary fields and derives the basic consequences of conformal invariance on quantum field theories, including the form of correlation functions and the Ward identities, then we discuss the representation theory of the Virasoro algebra. We study conformal field theories on torus where new consistency conditions arise from the action of the modular group. We present some simple but important example such as the free fermions, the free bosons. In follow, we study infinitesimal of Lie algebras known Kac-Moody algebra and we see how they define a presentation of the Sugawara and coset constructions. Then, we calculate the density of states in exponential limit. Key words: conformal field theory, primary fields, Virasoro algebra, Kac-Moody algebra
