  We give a survey of the background and recent development on the
$K$-equivalence relation among birational manifolds. After a brief historical
sketch of birational geometry, we define the $K$-partial ordering and
$K$-equivalence in a birational class and discuss geometric situations that
lead to these notions. One application to the filling-in problem for threefolds
is given.
  We discuss the motivic aspect of $K$-equivalence relation. We believe that
$K$-equivalent manifolds have the same Chow motive though we are unable to
prove it at this moment. Instead we discuss various approaches toward the
corresponding statements in different cohomological realizations.
  We also formulate the {\it Main Conjectures} and prove a weak version of it.
Namely, up to complex cobordism, $K$-equivalence can be decomposed into
composite of classical flops. Finally we review some other current researches
that are related to the study of $K$-equivalence relation.
