The discovery of binary pulsar PSR 1913+16 by Hulse & Taylor in 1974 established the existence of gravitational waves, for which the 1983 Nobel Prize was awarded. However, the measurement of astrophysical parameters from gravitational waves will open an entirely new spectrum for discovery and understanding of the Universe, not simply a new window in the electromagnetic spectrum like gamma ray telescopes in the 1970s. Two types of ground-based detectors, Advanced LIGO/Virgo and Pulsar Timing Arrays, are expected to directly detect gravitational waves in their respective frequency bands before the end of the decade. However, many of the most exciting sources are in the band from 0.1–100 mHz, accessible only from space due to seismic and gravity gradient noise on Earth. The European Space Agency (ESA) has chosen the 'Gravitational Universe' as the science theme for its L3 Cosmic Visions opportunity, planned for launch in 2034. NASA is planning to participate as a junior partner. Here we summarize progress toward realizing a gravitational wave observatory in space.