The holographic principle is a way to solve the problem of gravitational quantization. According to this hypothesis, we can describe any event that happens inside a volume by using a theorem that has been written on the surface. In this thesis, we explain the duality between a conformal field theory in two dimensional spaces and anti de Sitter 2+1 dimensional space. This explanation is a good evidence for this hypothesis. At first, we investigate the conformal field theory in 2-dimension, by representing its formalization and its important subjects. The procedure is based on some constraints that we put on correlation functions, according to symmetries. We do this work by introduction of energy momentum tensor, central charge, the mode expansion and the Hilbert space of our theory. we obtain these quantities for free fermionic and free bosonic theory. After that, we focus on the free theorem on torus, by using Hamiltonian interpretation. As an example we apply the fermionic theory to that. After CFT 2 survey, we should elucidate anti de Sitter space in 2+1 dimension. In this part, we clarify the solution for BTZ black hole and its metric characters. This metric is the solution of Einstein field equations with negative cosmological constant. it have time and angularIn BTZ metric, there is a singularity in causal structure and we will show this by geodesic analysis of stable black hole. In continuum, we solve this problem by regularization of metric. Then we pay attention to conformal coupled scalar field quantization in BTZ background. By image method, we obtain Green function for rotational and stable black holes. In extremal case we find out Green function by limitation process from the Green function of rotational black hole. Then we calculate the quantum energy momentum tensor for those states. For static, rotating and extreme black holes the energy tensor on the event horizon has a finite value, but in singular point will diverge. At last, we consider duality of AdS 3 /CFT 2 . This has been done by introduction of asymptotic symmetry for AdS 3 and this fact that the algebra between the generators of this symmetry is isomorphic to asymptotic symmetry generators of CFT 2 .The conformal weight of dual operator for a massive scalar field, conformal coupled scalar field and two point function bulk to boundary have been represented. Keywords : Conformal Field Theory, CFT, Primary Field, Conformal Weight, Energy Momentum Tensor, Virasoro Algebra, Central Charge, Radial Quantization, BTZ Black Hole,,Anti de Sitter Space, ADS, Identification, Geodesic, Extremal Black Hole, Green Function, Two Point Function, Scalar Field, Holographic Principle, Asymptotic Symmetry, Duality AdS/CFT, Einstein Gravity, Duality Gravity/CFT.