$L_\infty$ and $A_\infty$ structures: then and now

Abstract

Looking back over 55 years of higher homotopy structures, I reminisce as I recall the early days and ponder how they developed and how I now see them. From the history of $A_\infty$-structures and later of $L_\infty$ -structures and their progeny, I hope to highlight some old results which seem not to have garnered the attention they deserve as well as some tantalizing new connections.