I believe that the papers on $A_\infty$-structures by Bernhard Keller are extremely well written: they provide the reader with an overview of the state of the art of research in the topic(s), applications and even understandable proofs. I suggest in particular

[*A-infinity algebras, modules and functor categories*][1]


  [1]: https://www.imj-prg.fr/~bernhard.keller/publ/ainffun.pdf