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Let X, Y be Banach spaces and let (X,Y) be the space of bounded linear operators from X to Y. We develop the theory of double operator integrals on (X,Y) and apply this theory to obtain commutator estimates of the form $\left|\right|f\left(B\right)S-{Sf\left(A\right)\left|\right|}_{(X,Y)}\le const\left|\right|BS-{SA\left|\right|}_{(X,Y)}$ for a large class of functions f, where A ∈ (X), B ∈ (Y) are scalar type operators and S ∈ (X,Y). In particular, we establish this estimate for f(t): = |t| and for diagonalizable operators on $X={\ell }_p$ and $Y={\ell }_q$ for p < q. We also study the estimate above in the setting of Banach ideals...