  Archdeacon (1987) proved that graphs embeddable on a fixed surface can be
$3$-coloured so that each colour class induces a subgraph of bounded maximum
degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no
$K_{t+1}$-minor can be $t$-coloured so that each colour class induces a
subgraph of bounded maximum degree. We prove a common generalisation of these
theorems with a weaker assumption about excluded subgraphs. This result leads
to new defective colouring results for several graph classes, including graphs
with linear crossing number, graphs with given thickness (with relevance to the
earth-moon problem), graphs with given stack- or queue-number, linklessly or
knotlessly embeddable graphs, graphs with given Colin de Verdi\`ere parameter,
and graphs excluding a complete bipartite graph as a topological minor.
