  In 1981 W. Fulton and R. MacPherson introduced the notion of bivariant theory
(BT), which is a sophisticated unification of covariant theories and
contravariant theories. This is for the study of singular spaces. In 2001 M.
Levine and F. Morel introduced the notion of algebraic cobordism, which is a
universal oriented Borel-Moore functor with products (OBMF) of geometric type,
in an attempt to understand better V. Voevodsky's (higher) algebraic cobordism.
In this paper we introduce a notion of oriented bivariant theory (OBT), a
special case of which is nothing but the oriented Borel-Moore functor with
products. The present paper is a first one of the series to try to understand
Levine-Morel's algebraic cobordism from a bivariant-theoretical viewpoint, and
its first step is to introduce OBT as a unification of BT and OBMF.
