Operator-valued Fourier multipliers are used to study well-posedness of integro-differential equations in Banach spaces. Both strong and mild periodic solutions are considered. Strong well-posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems. We are concerned with a large array of vector-valued function spaces: Lebesgue-Bochner spaces L-p, the Besov spaces B-p,q(s) (and related spaces such as the Holder-Zygmund spaces C-s) and the Triebel-Lizorkin spaces F-p,q(s). We note that the Multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary Banach spaces. The results are applied to various classes of nonlinear integral and integro-differential equations. (C) 2008 Elsevier Inc. All rights reserved.