  We prove a rainbow version of the blow-up lemma of Koml\'os, S\'ark\"ozy and
Szemer\'edi for $\mu n$-bounded edge colourings. This enables the systematic
study of rainbow embeddings of bounded degree spanning subgraphs. As one
application, we show how our blow-up lemma can be used to transfer the
bandwidth theorem of B\"ottcher, Schacht and Taraz to the rainbow setting. It
can also be employed as a tool beyond the setting of $\mu n$-bounded edge
colourings. Kim, K\"uhn, Kupavskii and Osthus exploit this to prove several
rainbow decomposition results. Our proof methods include the strategy of an
alternative proof of the blow-up lemma given by R\"odl and Ruci\'nski, the
switching method, and the partial resampling algorithm developed by Harris and
Srinivasan.
