Orthogonal polynomials, Toda lattices and Painlevé equations

We give a survey of the connection between orthogonal polynomials, Toda lattices and related lattices, and Painlevé equations (discrete and continuous). •Derivation of the Toda lattice equations using orthogonal polynomials on the real line.•Derivation of the Ablowitz–Ladik equations using orthogonal polynomials on the unit circle.•Discrete Painlevé equations for the recurrence coefficients of orthogonal polynomials.•Painlevé differential equations follow by combining Toda lattice equations and discrete Painlevé equations.