Summary: Koszul and homogeneous Koszul algebras were defined by S. B. Priddy [Trans. Am. Math. Soc. 152, 39–60 (1970; Zbl 0261.18016)]. There exist some relations between these algebras and the skew PBW extensions introduced in [C. Gallego and O. Lezama, Commun. Algebra 39, No. 1, 50–75 (2011; Zbl 1259.16053)]. 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.