Conformal Field Theory (CFT) is a Quantum Field Theory, which is invariant under conformal transformations. In this thesis, we study the upper bound on the scaling dimensions of the lowest primary operator (other than the identity) in a CFT 2 . We first review the conformal field theory in two dimensions, by introducing primary fields, the operator product expansion, stress-energy tensor and the central charge. We review the virasoro algebra as the symmetry algebra of this theory. We then discussing the unitary representations of the Virasoro algebra. We study Conformal field theory on a torus, focusing on the modular invariance of the partition function for free fermions. Using modular invariance of the partition function, in the saddle point approximation. We obtain the asymptotic behavior of the density of states in high energies. The following, we consider CFT 2 with the central charge greater than one, which do not have any chiral algebra beyond the Virasoro algebra. Using modular invariance of the partition function at the medium-temperature, we obtain a constraint on a partition function. This constraint helps us to derive an upper bound on the conformal dimensions of the primary fields. We review different approaches to determining the upper bound on the scaling dimension of the lowest primary field (other than the identity) and we see that the leading term of the upper bound is equal to the one-twelfth of the central charge.