Koszul and homogeneous Koszul algebras were defined by Priddy in [18]. There exist some relations between these algebras and the skew PBW extensions introduced in [7]. In this paper, we give conditions to guarantee that skew PBW extensions over fields are Koszul or homogeneous Koszul. We also show that a constant skew PBW extension of a field is a PBW deformation of its homogeneous version.

skew PBW extensions, pre-Koszul algebras, Koszul algebras, PBW algebras, PBW deformations.