  The purpose of this survey article is a comprehensive study of operator
Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is
called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary
self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary
conditions for operator Lipschitzness. We also study the class of operator
differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz
functions on closed subsets of the plane as well as commutator Lipschitz
functions on such subsets. Am important role is played by double operator
integrals and Schur multipliers.
