These notes, based on lectures given at the Fifth Mexican School of Particles and Fields, provide an elementary introduction to quantum symmetries from a physical perspective. The focus is on integrable two-dimensional systems, which have an infinite number of conserved charges. The notes cover relativistic dynamics in one spatial dimension, Bethe ansatz, integrable vertex models, Yang-Baxter algebras, and the quantum group $U_q(\mathcal{G})$. The Yang-Baxter equation, a key feature of integrable systems, is discussed in detail, along with its implications for the structure of these systems. The notes also explore the connection between the Bethe ansatz and the diagonalization of spin chain Hamiltonians, and the application of these concepts to the six-vertex model, a classical statistical system. The Yang-Baxter algebra is introduced as a general framework to capture the integrability properties of the models studied.These notes, based on lectures given at the Fifth Mexican School of Particles and Fields, provide an elementary introduction to quantum symmetries from a physical perspective. The focus is on integrable two-dimensional systems, which have an infinite number of conserved charges. The notes cover relativistic dynamics in one spatial dimension, Bethe ansatz, integrable vertex models, Yang-Baxter algebras, and the quantum group $U_q(\mathcal{G})$. The Yang-Baxter equation, a key feature of integrable systems, is discussed in detail, along with its implications for the structure of these systems. The notes also explore the connection between the Bethe ansatz and the diagonalization of spin chain Hamiltonians, and the application of these concepts to the six-vertex model, a classical statistical system. The Yang-Baxter algebra is introduced as a general framework to capture the integrability properties of the models studied.