  Hadwiger's Conjecture asserts that every $K_h$-minor-free graph is properly
$(h-1)$-colourable. We prove the following improper analogue of Hadwiger's
Conjecture: for fixed $h$, every $K_h$-minor-free graph is $(h-1)$-colourable
with monochromatic components of bounded size. The number of colours is best
possible regardless of the size of monochromatic components. It solves an open
problem of Edwards, Kang, Kim, Oum and Seymour [\emph{SIAM J. Disc. Math.}
2015], and concludes a line of research initiated in 2007. Similarly, for fixed
$t\geq s$, we show that every $K_{s,t}$-minor-free graph is $(s+1)$-colourable
with monochromatic components of bounded size. The number of colours is best
possible, solving an open problem of van de Heuvel and Wood [\emph{J.~London
Math.\ Soc.} 2018]. We actually prove a single theorem from which both of the
above results are immediate corollaries. For an excluded apex minor, we
strengthen the result as follows: for fixed $t\geq s\geq 3$, and for any fixed
apex graph $X$, every $K_{s,t}$-subgraph-free $X$-minor-free graph is
$(s+1)$-colourable with monochromatic components of bounded size. The number of
colours is again best possible.
