  In these lecture notes we aim for a pedagogical introduction to both
classical and quantum integrability. Starting from Liouville integrability and
passing through Lax pair and r-matrix we discuss the construction of the
conserved charges for classical integrable models taking as example the
harmonic oscillator. The construction of these charges for 2D integrable field
theories is also discussed using a Lax connection and the Sine-Gordon model as
example. On the quantum side, the XXZ spin chain is used to explain the
systematic construction of the conserved charges starting from a quantum
R-matrix, solution of the quantum Yang-Baxter equation. The diagonalization of
these charges is performed using the algebraic Bethe ansatz. At the end, the
interpretation of the R-matrix as an S-matrix in a scattering process is also
presented. These notes were written for the lectures delivered at the school
"Integrability, Dualities and Deformations", that ran from 23 to 27 August 2021
in Santiago de Compostela and virtually.
