  Hadwiger's Conjecture asserts that every $K_t$-minor-free graph has a proper
$(t-1)$-colouring. We relax the conclusion in Hadwiger's Conjecture via
improper colourings. We prove that every $K_t$-minor-free graph is
$(2t-2)$-colourable with monochromatic components of order at most
$\lceil{\frac12(t-2)}\rceil$. This result has no more colours and much smaller
monochromatic components than all previous results in this direction. We then
prove that every $K_t$-minor-free graph is $(t-1)$-colourable with
monochromatic degree at most $t-2$. This is the best known degree bound for
such a result. Both these theorems are based on a decomposition method of
independent interest. We give analogous results for $K_{s,t}$-minor-free
graphs, which lead to improved bounds on generalised colouring numbers for
these classes. Finally, we prove that graphs containing no $K_t$-immersion are
$2$-colourable with bounded monochromatic degree.
