Abstract: Let X be a variety over a field of characteristic 0. Given a vector bundle E on X we construct Chern forms c i ( E ;∇) ∈ Γ( X ,A 2 i X ). Here A · X is the sheaf of Beilinson adeles and ∇ is an adelic connection . When X is smooth H p Γ( X ,A · X ) = H p DR ( X ), the algebraic De Rham cohomology, and c i ( E ) = [ c i ( E ;∇)] are the usual Chern classes. We include three applications of the construction: (1) existence of adelic secondary (Chern-Simons) characteristic classes on any smooth X and any vector bundle E ; (2) proof of the Bott Residue Formula for a vector field action; and (3) proof of a Gauss-Bonnet Formula on the level of differential forms, namely in the De Rham-residue complex.