  We prove a structural characterization of graphs that forbid a fixed graph
$H$ as an immersion and can be embedded in a surface of Euler genus $\gamma$.
In particular, we prove that a graph $G$ that excludes some connected graph $H$
as an immersion and is embedded in a surface of Euler genus $\gamma$ has either
"small" treewidth (bounded by a function of $H$ and $\gamma$) or "small" edge
connectivity (bounded by the maximum degree of $H$). Using the same techniques
we also prove an excluded grid theorem on bounded genus graphs for the
immersion relation.
