Immunological memory is considered to be one of the cardinal features of the adaptive immune system. Despite being a recognized phenomenon since the time of the ancient Greeks, immunologists are yet to fully appreciate the mechanisms that control memory responses in the immune system. Furthermore, our definition of immunological memory itself continues to evolve, with recent suggestions that innate immune cells also show memory-like behaviour. In this Viewpoint article, Nature Reviews Immunology invites five leading immunologists to share their thoughts on our current understanding of the nature of immunological memory. Our experts highlight some of the seminal studies that have shaped the immune memory field and offer contrasting views on the key questions that remain to be addressed.