We prove that for every d at least 2, deciding if a pure, d–dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every d at least 2 and k at least 0, deciding if a pure, d–dimensional, simplicial complex is k–decomposable is NP-hard. For d at least 3, both problems remain NP-hard when restricted to contractible pure d–dimensional complexes.