  An old and well-known conjecture of Frankl and F\"{u}redi states that the
Lagrangian of an $r$-uniform hypergraph with $m$ edges is maximised by an
initial segment of colex. In this paper we disprove this conjecture by finding
an infinite family of counterexamples for all $r \ge 4$. We also show that, for
sufficiently large $t \in \mathbb{N}$, the conjecture is true in the range
$\binom{t}{r} \le m \le \binom{t+1}{r} - \binom{t-1}{r-2}$.
