This paper describes a number of potential observing programs for Darwin in the area of galactic nuclei. Observations of our Galactic centre at the shorter wavelengths (5 micron ) covered by Darwin will detect large numbers of stars. Extrapolation of the existing K band counts to the flux levels expected to be reached by Darwin in 1 hour suggests 5,000 detectable stars (arc sec)-2 around the compact radio source Sgr A*, which is commonly believed to be associated with a black hole of mass 2.6 million M_o. Over its lifetime, Darwin will be capable of exquisitely accurate measurements of stellar proper motions, including the ability to follow entire stellar orbits around the hole for stars with orbital radii 10-3 pc. It may also be able to observe the general relativistic advance of the pericenter of stellar orbits somewhat closer to the hole. Stars behind the hole will be gravitationally lensed by it. It appears that the probability of observing a lensing event at any given time is of order unity at Darwin's resolution and sensitivity, so this should be a rich area of research. Darwin will measure the infrared spectrum of Sgr A* with a gain of 4 orders of magnitude of sensitivity over present upper limits, easily detecting the source if it conforms to the predictions of current ADAF models. Stars which approach too close to the black hole will be tidally disrupted by it, and Darwin's resolution is sufficiently good to spatially separate such a tidally disrupted red giant or supergiant from the black hole itself. However, the event rate is almost certainly too low for such a disruption to be seen in the mission lifetime. Dust shells around stars in the vicinity of the Galactic centre will be promising candidates for study. Darwin will be able to study the stellar luminosity function and stellar proper motions in the nucleus of M31. The possibility of using the stellar proper motions to estimate the mass of the nuclear black hole is discussed. Potential investigations of the distributions of warm dust and gas in the nuclei of nearby active galaxies are described. Darwin should be able to image the dusty accretion disks in Seyfert galaxies, compact starbursts around active nuclei and, depending on the available spectral resolution, study the stratification of different ionic species through observations of fine structure lines. It may be possible to resolve the broad line region if the wavelength coverage extends to short enough wavelengths. Darwin's nulling mode will be valuable for blocking the bright, compact light of quasars and investigating the kinematics of the surrounding gas. Whether stars, gas or dust are observed, galactic nuclei present densely-populated, bright fields, so the primary technical requirement for good imaging fidelity is very extensive coverage of the uv plane.