Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold $\mathcal{X}$ which depend in addition on a vector bundle over $\mathcal{X}$ and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle and to genus-zero one-point invariants of complete intersections in $\mathcal{X}$ . We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a “quantum Lefschetz theorem” that expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold $\mathcal{X}$ . We determine the genus-zero Gromov-Witten potential of the type $A$ surface singularity $[\mathbb{C}^2/\mathbb{Z}_n]$ . We also compute some genus-zero invariants of $[\mathbb{C}^3/\mathbb{Z}_3]$ , verifying predictions of Aganagic, Bouchard, and Klemm [3]. In a self-contained appendix, we determine the relationship between the quantum cohomology of the $A_n$ surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and of Bryan and Graber [12] in this case