This monograph presents a general construction of characteristic currents for singular connections on a vector bundle. It develops, in particular, a Chern-Weil theory for smooth bundle maps alpha: "formule mathématique non exprimable en texte html" for smooth connections on E and F, establishes formulas of the type

Here "formule mathématique non-exprimable en html" is a standard characteristic form, Res "formule mathématique non exprimable en texte html" is an associated smooth «residue» form computed canonically in terms of curvature Sigma_alpha is a rectifiable current depending only on the singular structure of alpha, and T is a canonical, functorial transgression form with coefficients in L¹_loc. The theory encompasses such classical topics as: Poincaré-Lelong Theory, Bott-Chern Theory, Chern-Weil Theory, and formulas of Hopf. Applications include: A new proof of the Riemann-Roch Theorem for vector bundles over algebraic curves; A C^infini generalization of the Poincaré-Lelong Formula; Universal formulas for the Thom class as an equivariant characteristic form (i.e., canonical formulas for a de Rham representative of the Thom class of a bundle with connection); A Differentiable Riemann-Roch-Grothendieck Theorem at the level of forms and currents. A variety of formulas relating geometry and characteristic classes are deduced as direct consequences of the theory.