It has been understood that quantum spacetime may be non-geometric in the sense
that its phase space algebra is noncommutative and nonassociative. It has therefore
been of interest to develop a formalism to describe differential geometry on
non-geometric spaces. Many of these spaces would fi t naturally as commutative algebra
objects in representation categories of triangular quasi-Hopf algebras because
they arise as cochain twist deformations of classical manifolds. In this thesis we develop
in a systematic fashion a description of differential geometry on commutative
algebra objects internal to the representation category of an arbitrary triangular
quasi-Hopf algebra. We show how to express well known geometrical concepts such
as tensor fields, differential calculi, connections and curvatures in terms of internal
homomorphisms using universal categorical constructions in a closed braided
monoidal category to capture algebraic properties such as Leibniz rules. This internal
description is an invaluable perspective for physics enabling one to construct
geometrical quantities with dynamical content and con guration spaces as large as
those in the corresponding classical theories. We also provide morphisms which lift
connections to tensor products and tensor elds. Working in the simplest setting
of trivial vector bundles we obtain explicit expressions for connections and their
curvatures on noncommutative and nonassociative vector bundles. We demonstrate
how to apply our formalism to the construction of a physically viable action functional
for Einstein-Cartan gravity on noncommutative and nonassociative spaces as
a step towards understanding the effect of nonassociative deformations of spacetime
geometry on models of quantum gravity
