  In this paper, we develop an enhancement of derived algebraic geometry to
apply to $\mathbb{A}^1$-homotopy theory introduced by Morel and Voevodsky. We
call the enhancement "motivic derived algebraic geometry". We shall actually
formulate "motivic" versions of $\infty$-categories, $\infty$-topoi, spectral
schemes and spectral Deligne--Mumford stacks established by Joyal, Lurie,
To\"en and Vezzosi.
  By using the language of motivic derived algebraic geometry, we construct the
Grassmannian and the algebraic $K$-theory. Furthermore we formulate the Thom
spaces for vector bundles on (motivic) stacks, and we obtain the algebraic
cobordism for (motivic) stacks. As the main result, we prove that the algebraic
cobordism corepresents the motivic $\infty$-category which has the universal
property of oriented (motivic) $\infty$-categories.
