  We define higher quantum Airy structures as generalizations of the
Kontsevich-Soibelman quantum Airy structures by allowing differential operators
of arbitrary order (instead of only quadratic). We construct many classes of
examples of higher quantum Airy structures as modules of
$\mathcal{W}(\mathfrak{g})$ algebras at self-dual level, with $\mathfrak{g}=
\mathfrak{gl}_{N+1}$, $\mathfrak{so}_{2 N }$ or $\mathfrak{e}_N$. We discuss
their enumerative geometric meaning in the context of (open and closed)
intersection theory of the moduli space of curves and its variants. Some of
these $\mathcal{W}$ constraints have already appeared in the literature, but we
find many new ones. For $\mathfrak{gl}_{N+1}$ our result hinges on the
description of previously unnoticed Lie subalgebras of the algebra of modes. As
a consequence, we obtain a simple characterization of the spectral curves (with
arbitrary ramification) for which the Bouchard-Eynard topological recursion
gives symmetric $\omega_{g,n}$s and is thus well defined. For all such cases,
we show that the topological recursion is equivalent to
$\mathcal{W}(\mathfrak{gl})$ constraints realized as higher quantum Airy
structures, and obtain a Givental-like decomposition for the corresponding
partition functions.
