A multiple-contact woven connector comprising at least one conductor woven onto a plurality of loading fibers. Each conductor has at least one contact point. The loading fibers are capable of delivering a contact force at the contact points of the conductors. The connector may further include mating conductors having a contact mating surfaces. Electrical connections can be established between the contact points of the conductors and the contact mating surfaces of the mating conductors. The contact mating surfaces may be curved. In exemplary embodiments, the contact mating surfaces are convex. The multiple-contact woven connectors of the present disclosure can be utilized as power connectors or, alternatively, as data connectors.