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A Clifford module bundle is a vector bundle over a Riemannian manifold with a fibrewise Clifford multiplication. A compatible connection on a vector bundle satisfies $[{\nabla }_X,c(\alpha )]=c({\nabla }_X\alpha )$ for the Levi-Civita connection on covectors. On a graded Clifford module bundle, Clifford multiplication exchanges the two grading summands. The associated Dirac operator is $D={\sum }_{i}c(e^i){\nabla }_{e_i}$. Its fibres are modules over the fibres of the Clifford algebra bundle.