What is higher Lie theory?

Lie theory" refers to a functorial correspondence between
finite-dimensional Lie algebras and 1-connected Lie groups, elucidated
by Sophus Lie and summarized in his three theorems, Lie I, II and III.
The first hint that there might be more to the story came from van Est
who noticed (and proved) that integrating higher Lie algebra cocycles
required higher connectivity assumptions on the Lie group. Furthermore,
attempts to build a Lie theory for Lie algebroids (and
infinite-dimensional Lie algebras) generally run into obstructions of a
homotopy-theoretic nature, leading to higher structures -- Lie 2-groups
and groupoids. At the same time, it was discovered that many geometric
structures arising in theoretical physics exhibit a hierarchy of
higher-order symmetries (symmetries between symmetries, and so on) which
behave a lot like homotopy types in algebraic topology. The goal of
higher Lie theory is to construct a higher-categorical version of Lie's
correspondence for "spaces with infinitesimal higher symmetries"
(modelled by differential graded manifolds) on the one hand, and "spaces
with global higher symmetries" (modelled by simplicial manifolds) on the
other. In this talk we will explain the problem and the main concepts
and constructions involved, and survey the progress so far.