  Let A be a cosemisimple Hopf *-algebra with antipode S and let $\Gamma$ be a
left-covariant first order differential *-calculus over A such that $\Gamma$ is
self-dual and invariant under the Hopf algebra automorphism S^2. A quantum
Clifford algebra $\Cl(\Gamma,\sigma,g)$ is introduced which acts on Woronowicz'
external algebra $\Gamma^\wedge$. A minimal left ideal of
$\Cl(\Gamma,\sigma,g)$ which is an A-bimodule is called a spinor module.
Metrics on spinor modules are investigated. The usual notion of a linear left
connection on $\Gamma$ is extended to quantum Clifford algebras and also to
spinor modules. The corresponding Dirac operator and connection Laplacian are
defined. For the quantum group SL_q(2) and its bicovariant $4D_\pm$-calculi
these concepts are studied in detail. A generalization of Bochner's theorem is
given. All invariant differential operators over a given spinor module are
determined. The eigenvalues of the Dirac operator are computed.
  Keywords: quantum groups, covariant differential calculus, spin geometry
