We solve the long-standing problem of Deligne-Simpson: given conjugacy classes $(C_j)_{1\le j\le k}$ of invertible matrices of rank $n$, do there exist $A_j\in C_j$ such that (1) $A_1\cdots A_k=\Id$ and (2) there is no nontrivial proper subspace of $\mathbb{C}^n$ that is preserved by every $A_j$? A conjectural necessary and sufficient condition on $(C_j)_j$ in terms of certain Kac-Moody root systems was proposed by Crawley-Boevey, and the sufficiency statement was later proved in his joint work with Shaw. Our main result proves the necessity statement and the method is a combination of nonabelian Hodge theory and variation of stability conditions.