Abstract: In this paper we present a Clifford bundle approach to the geometry of a general Riemann-Cartan-Weyl space (RCWS). In our formulation, the so-called Dirac operators play a funda- mental role. We first introduce one these operators in the context of a Riemannian space, calling it the fundamental Dirac Dirac operator (8). We show, among other important results, that the fundamental Dirac operator can be written =d-6, where d and 6 are the familiar exterior derivative operator and the Hodge codifferential acting on sections of the Hodge bundle (interpreted as embedded in the Clifford bundle). In connection with the fundamental Dirac operator, we introduce the concepts of Dirac commutator and anticommutator and we investigate their geometrical interpretation. With the theory of the sym- metric automorphisms’ s of a Clifford algebra we introduce infinitely many other Dirac-like operators, one for each nondegenerate bilinear form field that can be defined on the metric manifold M. We introduce also the concepts of Ricci and Einstein operators which are useful for intrinsic formulations of Einstein's gravitational theory. Later we generalize the fundamental Dirac operator and the Dirac commutators to a RCWS, showing their relations. In particular, we succeed in finding the correct generalization of the Hodge Laplacian to Riemann-Cartan spaces and we identify the natural wave operator (£+) for these spaces. Apart from a constant factor, £+ is the relativistic Hamiltonian operator which generates the theory of Markov processes of the Stochastic Mechanics. We obtain also new decompositions of the general affine connection V defining the RCWS structure and we identify new tensor objects. Our findings clear many results obtained in formulations of the flat space theory of gravitational field and of the theory of spinor fields in RCWS and suggest several generalizations of these theories. This subject is discussed in two following papers Ver menos