  The topic of this thesis is the development of a versatile and geometrically
motivated differential calculus on non-commutative or quantum spaces, providing
powerful but easy-to-use mathematical tools for applications in physics and
related sciences. A generalization of unitary time evolution is proposed and
studied for a simple 2-level system, leading to non-conservation of microscopic
entropy, a phenomenon new to quantum mechanics. A Cartan calculus that combines
functions, forms, Lie derivatives and inner derivations along general vector
fields into one big algebra is constructed for quantum groups and then extended
to quantum planes. The construction of a tangent bundle on a quantum group
manifold and an BRST type approach to quantum group gauge theory are given as
further examples of applications. The material is organized in two parts: Part
I studies vector fields on quantum groups, emphasizing Hopf algebraic
structures, but also introducing a `quantum geometric' construction. Using a
generalized semi-direct product construction we combine the dual Hopf algebras
\A\ of functions and \U\ of left-invariant vector fields into one fully
bicovariant algebra of differential operators. The pure braid group is
introduced as the commutant of $\Delta (\U)$. It provides invariant maps $\A
\to \U$ and thereby bicovariant vector fields, casimirs and metrics. This
construction allows the translation of undeformed matrix expressions into their
less obvious quantum algebraic counter parts. We study this in detail for
quasitriangular Hopf algebras, giving the determinant and
