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 $\alpha : E\to F$ which, for smooth connections on $E$ and $F$, establishes formulas of the type $\phi = \mathrm {Res}_\phi \Sigma _\alpha + dT.$ Here $\phi $ is a standard characteristic form, Res$_\phi $ 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 $L^1_{loc}$. The theory encompasses such ical 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^\infty $-generalization of the Poincaré-Lelong Formula ; Universal formulas for the Thom as an equivariant characteristic form (i. e. , canonical formulas for a de Rham representative of the Thom 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 es are deduced as direct consequences of the theory.