  In this thesis we present some computations made in both sides of the AdS/CFT
holographic correspondence using the integrability of both theories.
  Regarding the string theory side, this thesis is focused in the computation
of the dispersion relation of closed spinning strings in some deformed $AdS_3
\times S^3$ backgrounds. In particular we are going to focus in the deformation
provided by the mixing of R-R and NS-NS fluxes and the so-called
$\eta$-deformation. These computations are made using the classical
integrability of these two deformed string theories, which is provided by the
presence of a set of conserved quantities called "Uhlenbeck constants". The
existence of the Uhlenbeck constants is central for the method used to derive
the dispersion relations.
  Regarding the gauge theory side, we are interested in the computation of two
and three-point correlation functions. Concerning the two-point function a
computation of correlation functions involving different operators and
different number of excitations is performed using the Algebraic Bethe Ansatz
and the Quantum Inverse Scattering Method. These results are compared with
computations done with the Coordinate Bethe Ansatz and with
Zamolodchikov-Faddeev operators. Concerning the three-point functions, we will
explore the novel construction given by the hexagon framework. In particular we
are going to start from the already proposed hexagon form factor and rewrite it
in a language more resembling of the Algebraic Bethe Ansatz. For this intent we
construct an invariant vertex using Zamolodchikov-Faddeev operators, which is
checked for some simple cases.
