AbstractWe examine the theory of connective algebraicK-theory,, defined by taking the −1 connective cover of algebraicK-theory with respect to Voevodsky's slice tower in the motivic stable homotopy category. We extendto a bi-graded oriented duality theorywhen the base scheme is the spectrum of a fieldkof characteristic zero. The homology theorymay be viewed as connective algebraicG-theory. We identifyforXa finite typek-scheme with the image ofin, whereis the abelian category of coherent sheaves onXwith support in dimension at mostn; this agrees with the (2n,n) part of the theory of connective algebraicK-theory defined by Cai. We also show that the classifying map from algebraic cobordism identifieswith the universal oriented Borel-Moore homology theoryhaving formal group lawu+υ−βuυwith coefficient ring ℤ[β]. As an application, we show that every pure dimensiondfinite typek-scheme has a well-defined fundamental class [X]CKin ΩdCK(X), and this fundamental class is functorial with respect to pull-back for l.c.i. morphisms.