  We develop a generalisation of the original theory of regularity structures,
[Hai14], which is able to treat SPDEs on manifolds with values in vector
bundles. Assume $M$ is a Riemannian manifold and $E\to M$ and $F^i\to M$ are
vector bundles (with a metric and connection), this theory allows to solve
subcritical equations of the form $$ \partial_t u + \mathcal{L}u = \sum_{i=0}^m
G_i(u, \nabla u,\ldots, \nabla^n u)\xi_i\ , $$ where $u$ is a (generalised)
section of $E$, $\mathcal{L}$ is a uniformly elliptic operator on $E$ of order
strictly greater than $n$, the $\xi_i$ are $F^i$-valued random distributions
(e.g. $F^i$-valued white noises), and the $G_i:E\times TM^*\otimes E
\times\ldots\times (TM^*)^{\otimes n} \otimes E \to L(F^i ,E)$ are local
functions.
  We apply our framework to three example equations which illustrate that when
$\mathcal{L}$ is a Laplacian it is possible in most cases to renormalise such
equations by adding spatially homogeneous counterterms and we discuss in which
cases more sophisticated renormalisation procedures (involving the curvature of
the underlying manifold) are required.
