We prove that every 0/1-polytope has a unique Minkowski decomposition into indecomposable polytopes, up to translation of summands. The summands lie in pairwise orthogonal subspaces. Thus, every 0/1-polytope is the Cartesian product of indecomposable 0/1-polytopes. As applications, we obtain uniform combinatorial indecomposability criteria for order and chain polytopes, matroid polytopes, stable set and clique polytopes, edge polytopes, flow polytopes, and 2-level/compressed polytopes. We also show that every nontrivial factorization of a multi-affine polynomial is a product of multi-affine polynomials in disjoint sets of variables.