  In this paper we prove the universal property of skew $PBW$ extensions
generalizing this way the well known universal property of skew polynomial
rings. For this, we will show first a result about the existence of this class
of non-commutative rings. Skew $PBW$ extensions include as particular examples
Weyl algebras, enveloping algebras of finite-dimensional Lie algebras (and its
quantization), Artamonov quantum polynomials, diffusion algebras, Manin algebra
of quantum matrices, among many others. As a corollary we will give a new short
proof of the Poincar\'e-Birkhoff-Witt theorem about the bases of enveloping
algebras of finite-dimensional Lie algebras.
