In this paper we initiate the study of enriched ∞-operads. We introduce several models for these objects, including enriched versions of Barwick's Segal operads and the dendroidal Segal spaces of Cisinski and Moerdijk, and show these are equivalent. Our main results are a version of Rezk's completion theorem for enriched ∞-operads: localization at the fully faithful and essentially surjective morphisms is given by the full subcategory of complete objects, and a rectification theorem: the homotopy theory of ∞-operads enriched in the ∞-category arising from a nice symmetric monoidal model category is equivalent to the homotopy theory of strictly enriched operads.