  We continue our investigation of the general notion of universal enveloping
algebra introduced in [A. Ardizzoni, \emph{A Milnor-Moore Type Theorem for
Primitively Generated Braided Bialgebras}, J. Algebra \textbf{327} (2011), no.
1, 337--365]. Namely we study when such an algebra is of PBW type, meaning that
a suitable PBW type theorem holds. We discuss the problem of finding a basis
for a universal enveloping algebra of PBW type: As an application we recover
the PBW basis both of an ordinary universal enveloping algebra and of a
restricted enveloping algebra. We prove that a universal enveloping algebra is
of PBW type if and only if it is cosymmetric. We characterize braided bialgebra
liftings of Nichols algebras as universal enveloping algebras of PBW type.
