  We extend the construction of the Hennings TQFT for ribbon Hopf algebras to
the case of ribbon quasi-Hopf algebras as defined by Drinfeld. Calculations
proceed in a similar fashion to the ordinary Hopf algebra case, but also
require the handling of the non-trivial coassociator in the triple tensor
product of the algebra as well as several special elements. The main technical
difficulties we encounter are representing tangle categories in the
non-associative setting, and the definition and use of integrals and
cointegrals in the non-coassociative case. We therefore discuss the integral
theory for quasi-Hopf algebras, using work of Hausser and Nill. A motivating
example for this work is the Dijkgraaf-Pasquier-Roche algebra which is believed
to be related to the Dijkgraaf-Witten TQFT.
