We use Lurie’s symmetric monoidal envelope functor to give two new descriptions of∞-operads: as certain symmetric monoidal ∞-categories whose underlying symmetric monoidal∞-groupoids are free, and as certain symmetric monoidal ∞-categories equipped with a symmetricmonoidal functor to finite sets (with disjoint union as tensor product). The latter leads to a thirddescription of ∞-operads, as a localization of a presheaf ∞-category, and we use this to give a simpleproof of the equivalence between Lurie’s and Barwick’s models for ∞-operads.2020 Mathemati