The Yang-Baxter-Zamolodchikov-Faddeev (YBZF) algebras and their many applications are the subject of this reivew. I start by the solvable lattice statistical models constructed from YBZF algebras. All two-dimensional integrable vertex models follow in this way and are solvable via Bethe Ansatz (BA) and their generalizations. The six-vertex model solution and its q(2q−1) vertex generalization including its nested BA construction are exposed. YBZF algebras and their associated physical models are classified in terms of simple Lie algebras. It is shown how these lattice models yield both solvable massive quantum field theories (QFT) and conformal models in appropriated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is described as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YBZF algebras. A discussion on braid and quantum groups concludes this review.
<!-- *** Custom HTML *** --> The Yang–Baxter algebras (YBA) are introduced and formulated in a general way stressing graphical methods. Their various physical applications are then exposed: lattice statistical models, integrable field theories and factorizable S-matrices. The Bethe Ansatz (BA) and its generalizations provide the explicit solutions of all these models using the appropiate YBA. The six-vertex model solution is exposed. YB algebras and their associated physical models are classified in terms of simple Lie algebras. It is exposed how these lattice models yield both solvable massive QFT and conformal models in appropiated scaling (continuous) limits within the lattice light-cone approach. The method of finite-size calculations from the BA is exposed as well as its applications to derive the conformal properties of integrable lattice models. It is conjectured that all integrable QFT and conformal models follow in a scaling limit from these YB algebras. To conclude braid and quantum groups are derived from the YBA in the limit of infinite spectral parameter.