We define the notion of an oriented cohomology theory on smooth, quasi-projective schemes over a field k, and construct, in the case k has characteristic zero, the universal oriented cohomology theory, which we call algebraic cobordism. We compute the algebraic cobordism of the base field k, and show that this is naturally isomorphic to the Lazard ring. We also verify a localization property for algebraic cobordism. Using these facts, we give a proof of Rost's degree formula and Rost's generalized degree formula. Finally, we relate the algebraic cobordism of X to the classical Chow ring of X and with the Grothendieck group of algebraic vector bundles on X.