  We study various problems arising in higher differential geometry using {\it
derived Lie $\infty$-groupoids and algebroids}.We first study Lie
$\infty$-groupoids in various categories of derived geometric objects in
differential geometry, including derived manifolds, derived analytic spaces,
derived noncommutative spaces, and derived Banach manifolds. We construct
category of fibrant objects (CFO) structures in the category of derived Lie
$\infty$-groupoids. Then we study $L_{\infty}$-algebroids which are the
infinitesimal counterpart of derived Lie $\infty$-groupoids. We then study the
homotopical algebras for derived Lie $\infty$-groupoids and algebroids and
study their homotopy-coherent representations, which we call
$\infty$-representations. We relate $\infty$-representations of
$L_{\infty}$-algebroids to (quasi-) cohesive modules developed by Block, and
$\infty$-representations of Lie $\infty$-groupoids to $\infty$-local system
introduced by Block-Smith. Then we apply these tools in studying singular
foliations and their characteristic classes. We construct Atiyah classes for
$L_{\infty}$-algebroids pairs. We study singular foliations and their
holonomies. We construct $\L_{\infty}$-algebroids for holomorphic singular
foliations, and then We study elliptic involutive structures and prove an
dg-enhancement of $V$-analytic coherent sheaves. These examples inspire us to
define {\it perfect singular foliations}, which is a subcategory of singular
foliation but with better homological algebras. Next, we construct various Lie
$\infty$-groupoids for singular foliations. Then we study foliations on stacks
and higher groupoids. Finally, we prove an $A_{\infty}$ de Rham theorem for
foliations, and Riemann-Hilbert correspondence for foliated $\infty$-local
system foliated manifolds.
