  In this paper we find big Euclidean domains in complex manifolds. We consider
open neighbourhoods of sets of the form $K\cup M$ in a complex manifold $X$,
where $K$ is a compact $\mathscr O(U)$-convex set in an open Stein
neighbourhood $U$ of $K$, $M$ is an embedded Stein submanifold of $X$, and
$K\cap M$ is compact and $\mathscr O(M)$-convex. We prove a Docquier-Grauert
type theorem concerning biholomorphic equivalence of neighbourhoods of such
sets, and we give sufficient conditions for the existence of Stein
neighbourhoods of $K\cup M$, biholomorphic to domains in $\mathbb C^n$ with
$n=\dim X$, such that $M$ is mapped onto a closed complex submanifold of
$\mathbb C^n$.
