  We prove that for every $d\geq 2$, 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 $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional,
simplicial complex is $k$-decomposable is NP-hard. For $d \ge 3$, both problems
remain NP-hard when restricted to contractible pure $d$-dimensional complexes.
Another simple corollary of our result is that it is NP-hard to decide whether
a given poset is CL-shellable.
