Summary: We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number \(k\) of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on \(k\)). This decomposition result parallels an analogous, former result for edge deletions instead of contractions, and it generalizes a similar previous result for “compression” (a variant of contraction) in planar graphs. Our decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction), a much more general class than minor-closed problems. We prove that any contraction-closed problem satisfying just a few simple conditions has a PTAS in bounded-genus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimum-weight \(c\)-edge-connected submultigraph on bounded-genus graphs, improving and generalizing several previous algorithms. We also highlight the only main difficulty in extending our results to general \(H\)-minor-free graphs.