The goal of this survey is a comprehensive study of operator Lipschitz functions. A continuous function f on the real line R is said to be operator Lipschitz if ||f(A) - f(B)|| 6 const||A - B|| for arbitrary self-adjoint operators A and B. Sufficient conditions and necessary conditions are given for operator Lipschitzness. The class of operator differentiable functions on R is also studied. Further, operator Lipschitz functions on closed subsets of the plane are considered, and the class of commutator Lipschitz functions on such subsets is introduced. An important role for the study of such classes of functions is played by double operator integrals and Schur multipliers.