Gravitational-wave (GW) memory leaves a lasting imprint on spacetime: a permanent offset in the GW strain that persists after a burst passes, producing a tiny but potentially measurable displacement of test masses. This characteristic low-frequency signal is a robust prediction of General Relativity, but it remains undetected. In this talk, I will explore the nonlinear GW memory from binary black hole mergers and the latest prospects for detecting it from supermassive black hole (SMBH) coalescences with LISA. I will then discuss recent applications of memory to GW signals at high frequency, in particular those from primordial black holes and from the early universe. Finally, I will outline how a detection of GW memory could provide a new and complementary test of General Relativity.