  We introduce the local and global indices of Dirac operators for the rational
Cherednik algebra $\mathsf{H}_{t,c}(G,\mathfrak{h})$, where $G$ is a complex
reflection group acting on a finite-dimensional vector space $\mathfrak{h}$. We
investigate precise relations between the (local) Dirac index of a simple
module in the category $\mathcal{O}$ of $\mathsf{H}_{t,c}(G,\mathfrak{h})$, the
graded $G$-character of the module, the Euler-Poincar\'e pairing, and the
composition series polynomials for standard modules. In the global theory, we
introduce integral-reflection modules for $\mathsf{H}_{t,c}(G,\mathfrak{h})$
constructed from finite-dimensional $G$-modules. We define and compute the
index of a Dirac operator on the integral-reflection module and show that the
index is, in a sense, independent of the parameter function $c$. The study of
the kernel of these global Dirac operators leads naturally to a notion of
dualised generalised Dunkl-Opdam operators.
