  In this article, we introduce and study a natural notion of coarse separation
for metric spaces, with an emphasis on coarse separation by subspaces of
polynomial or subexponential growth. For instance, we show that symmetric
spaces of non-compact type different from $\mathbb{H}_\mathbb{R}^2$ and thick
Euclidean buildings of rank $\geq 2$ cannot be coarsely separated by subspaces
of subexponential growth; and that a connected nilpotent Lie group of growth
degree $D \geq 2$ cannot be coarsely separated by a subspace of polynomial
degree $\leq D-2$. We apply these results to the large-scale geometry of
amalgamated free products and wreath products. The latter application is based
on an Embedding Theorem that generalises previous work of the last two authors,
and which is of independent interest. We also discuss some further applications
to the distorsion of coarse embeddings between certain metric spaces.
