Freely generated vertex algebras and non-linear Lie conformal algebras

We introduce the notion of a non--linear Lie conformal superalgebra and prove a PBW theorem for its universal enveloping vertex algebra. We also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a non--linear Lie conformal superalgebra. This correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras.