Let A : [0, τ] → L (D, X) be strongly measurable and bounded, where D, X are Banach spaces such that D {right arrow, hooked} X. We assume that the operator A (t) has maximal regularity for all t ∈ [0, τ]. Then we show under some additional hypothesis (viz. relative continuity) that the non-autonomous problem(P) over(u, ̇) + A (t) u = f a.e. on (0, τ), u (0) = x, is well-posed in Lp; i.e. for all f ∈ Lp (0, τ ; X) and all x ∈ (X, D)frac(1, p*), p there exists a unique u ∈ W1, p (0, τ ; X) ∩ Lp (0, τ ; D) solution of (P), where 1 < p < ∞. If the operators A (t) are accretive, we show that conversely, well-posedness of (P) implies that A (t) has maximal regularity for all t ∈ [0, τ]. We also consider the non-autonomous second order problemover(u, ̈) + B (t) over(u, ̇) + A (t) u = f a.e. on (0, τ), u (0) = x, over(u, ̇) (0) = y, for which we prove similar regularity and perturbation results. © 2007 Elsevier Inc. All rights reserved.