  The millihertz gravitational wave band can only be accessed with a
space-based interferometer, but it is one of the richest in potential sources.
Observations in this band have amazing scientific potential. The mergers
between massive black holes with mass in the range 10 thousand to 10 million
solar masses, which are expected to occur following the mergers of their host
galaxies, produce strong millihertz gravitational radiation. Observations of
these systems will trace the hierarchical assembly of structure in the Universe
in a mass range that is very difficult to probe electromagnetically. Stellar
mass compact objects falling into such black holes in the centres of galaxies
generate detectable gravitational radiation for several years prior to the
final plunge and merger with the central black hole. Measurements of these
systems offer an unprecedented opportunity to probe the predictions of general
relativity in the strong-field and dynamical regime. Millihertz gravitational
waves are also generated by millions of ultra-compact binaries in the Milky
Way, providing a new way to probe galactic stellar populations. ESA has
recognised this great scientific potential by selecting The Gravitational
Universe as its theme for the L3 large satellite mission, scheduled for launch
in ~2034. In this article we will review the likely sources for millihertz
gravitational wave detectors and describe the wide applications that
observations of these sources could have for astrophysics, cosmology and
fundamental physics.
