The Erdos-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on$n$vertices is at most n. Erdös considered this to be one of his three most favorite combinatorial problems and offered a $500 reward for a proof of this conjecture. We prove this conjecture for every large n. Here, we also provide a randomised algorithm to find such a colouring in polynomial time with high probability.