We study the unification problem for the theories of nilpotent groups of class at least 5. This is equivalent to the problem of constructing an algorithm which will solve any equation in a free nilpotent group of class at least 5. Romankov in 1977 showed that the ‘endomorphic reducibility’ problem is undecidable for free nilpotent groups of class at least 9 and it follows that the Unification Problem for the theories of nilpotent groups of class at least 9 is undecidable. We shall improve this to 5 by reducing the problem of algorithmically solving an arbitrary diophantine equation of degree 4 to that of solving equations in a free nilpotent group of class 5 and then appealing to Matiyasevitch's Theorem.