  The aim of this article is to extend the scope of the theory of regularity
structures in order to deal with a large class of singular SPDEs of the form
$$\partial_t u = \mathfrak{L} u+ F(u, \xi)\ ,$$ where the differential operator
$\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base
space $\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group $\mathbb{G}$ such
that the differential operator $\partial_t -\mathfrak{L}$ becomes a translation
invariant hypoelliptic operator on $\mathbb{G}$. Prime examples are the kinetic
Fokker-Planck operator $\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type
operators associated to sub-Laplacians. As an application of the developed
framework, we solve a class of parabolic Anderson type equations $$\partial_t u
= \sum_{i} X^2_i u + u (\xi-c)$$ on the compact quotient of an arbitrary Carnot
group.
