  For a linear algebraic group $G$ over a field $k$, we define an equivariant
version of the Voevodsky's motivic cobordism $MGL$. We show that this is an
oriented cohomology theory with localization sequence on the category of smooth
$G$-schemes and there is a natural transformation from this functor to the
functor of equivariant motivic cohomology. We give several applications. In
particular, we use this equivariant motivic cobordism to study the cobordism
ring of the classifying spaces and the cycle class maps from the algebraic to
the singular cohomology of such spaces. This theory of motivic cobordism allows
us to define the theory of motivic cobordism on the category of all smooth
quotient stacks.
