In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space 1 X over a Riemannian manifold X. This geometry is ``non-flat'' even if X=R d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient { 1 , divergence div 1 , and Laplace Beltrami operator H 1 =&div 1 { 1 are constructed. The associated volume elements, i.e., all measures + on 1 X w.r.t. which { 1 and div 1 become dual operators on L 2 (1 X ; +), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on 1 X . The corresponding Dirichlet forms E 1 + on L 2 (1 X ; +) are, therefore, defined. Each is shown to be associated with a diffusion process which is thus the Brownian motion on 1 X and which is subsequently identified as the usual independent infinite particle process on X. The associated heat semigroup (T 1 + (t)) t>0 is calculated explicitly. It is also proved that the diffusion process, when started with +, is time-ergodic (or equivalently E 1 + is irreducible or equivalently (T 1 + (t)) t>0 is ergodic) if and only if + is Poisson measure ? z dx with intensity z dx for some z 0. Furthermore, it is shown that the Laplace Beltrami operator H 1 =&div 1 { 1 on L 2 (1 X ; ? z dx ) is unitary equivalent to the second quantization of the Laplacian &2 X on X on the corresponding Fock space n 0 L 2 (X; z dx) n . As another direct consequence of our results we obtain a representation of the Lie-algebra of compactly supported vector fields on X on Poisson space. Finally, generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on X are described, in particular, the case where the mixed Poisson measures + are replaced by Gibbs measures of Ruelle-type on 1 X .