Abstract: Spacetime itself in form of gravitational waves is delivering information from faraway regions of the Universe.A new era began with the first detection of gravitational waves by Advanced LIGO in September 2015, and since then several events have been recorded by the LIGO/VIRGO collaboration. New challenges await us to unravel the delightful interplay between physics, astrophysics and mathematics. I will present new structures in various classes of spacetimes and their consequences for gravitational waves and their memory (a permanent change of the spacetime by gravitational waves). Gravitational wave memory has been predicted by General Relativity (GR). There are two types of memory, one going back to Ya. B. Zel'dovich and A. G. Polnarev and one to D. Christodoulou. The former is sourced by the change of a particular component of the curvature tensor the latter by fields that reach null infinity. With P. Chen, S.-T. Yau, and D. Garfinkle we showed that stress-energy that reaches null infinity also contributes to the null memory. Moreover, with D. Garfinkle we found two electromagnetic analogs of memory for the Maxwell equations. In order to understand the dynamics of the gravitational field we analyze classes of spacetimes, that is we solve the Cauchy problem (i.e. initial value problem) for the Einstein equations. I will also discuss my results on the parity of gravitational wave memory, and recent joint work with D. Garfinkle on borderline cases.