Summary: We generalize the structure theorem of Robertson and Seymour for graphs excluding a fixed graph \(H\) as a minor to graphs excluding \(H\) as a topological subgraph. We prove that for a fixed \(H\), every graph excluding \(H\) as a topological subgraph has a tree decomposition where each part is either “almost embeddable” to a fixed surface or has bounded degree with the exception of a bounded number of vertices. Furthermore, we prove that such a decomposition is computable by an algorithm that is fixed-parameter tractable with parameter \(|H|\). We present two algorithmic applications of our structure theorem. To illustrate the mechanics of a “typical” application of the structure theorem, we show that on graphs excluding \(H\) as a topological subgraph, partial dominating set (find \(k\) vertices whose closed neighborhood has maximum size) can be solved in time \(f(H,k)\cdot n^{O(1)}\). More significantly, we show that on graphs excluding \(H\) as a topological subgraph, Graph Isomorphism can be solved in time \(n^{f(H)}\). This result unifies and generalizes two previously known important polynomial time solvable cases of graph isomorphism: bounded-degree graphs [E. M. Luks, J. Comput. Syst. Sci. 25, 42–65 (1982; Zbl 0493.68064)] and \(H\)-minor free graphs [I. N. Ponomarenko, J. Sov. Math. 55, No. 2, 1621–1643 (1991); translation from Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 174, 147–177 (1988; Zbl 0745.05035)]. The proof of this result needs a generalization of our structure theorem to the context of invariant treelike decomposition.