  A subgraph of an edge-coloured graph is called rainbow if all its edges have
different colours. We prove a rainbow version of the blow-up lemma of Koml\'os,
S\'ark\"ozy and Szemer\'edi that applies to almost optimally bounded
colourings. A corollary of this is that there exists a rainbow copy of any
bounded-degree spanning subgraph $H$ in a quasirandom host graph $G$, assuming
that the edge-colouring of $G$ fulfills a boundedness condition that is
asymptotically best possible.
  This has many applications beyond rainbow colourings, for example to graph
decompositions, orthogonal double covers and graph labellings.
