  These notes are based on a series of three lectures given (online) by the
first named author at the workshop "Higher Structures and Operadic Calculus" at
CRM Barcelona in June 2021. The aim is to give a concise introduction to
rational homotopy theory through the lens of higher structures.
  The rational homotopy type of a simply connected space of finite type is
modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or
alternatively an $L_\infty$-algebra structure on the rational homotopy groups.
The first lecture is devoted to explaining these models and their relation to
the classical models of Quillen and Sullivan.
  The second lecture discusses the relation between Koszul algebras, formality
and coformality. The main result is that a space is formal if and only if the
rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal
if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are
both formal and coformal, this collapses to classical Koszul duality between
Lie and commutative algebras.
  In the third lecture, we discuss certain higher structure in the rational
homotopy theory of automorphisms of high dimensional manifolds, discovered by
Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph
complex and variants of it.
