  The $k$-colouring reconfiguration problem asks whether, for a given graph
$G$, two proper $k$-colourings $\alpha$ and $\beta$ of $G$, and a positive
integer $\ell$, there exists a sequence of at most $\ell+1$ proper
$k$-colourings of $G$ which starts with $\alpha$ and ends with $\beta$ and
where successive colourings in the sequence differ on exactly one vertex of
$G$. We give a complete picture of the parameterized complexity of the
$k$-colouring reconfiguration problem for each fixed $k$ when parameterized by
$\ell$. First we show that the $k$-colouring reconfiguration problem is
polynomial-time solvable for $k=3$, settling an open problem of Cereceda, van
den Heuvel and Johnson. Then, for all $k \geq 4$, we show that the
$k$-colouring reconfiguration problem, when parameterized by $\ell$, is
fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman,
Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial
hierarchy collapses.
