We present a model intercomparison project, LongRunMIP, the first collection of millennial-length (1,000+ years) simulations of complex coupled climate models with a representation of ocean, atmosphere, sea ice, and land surface, and their interactions. Standard model simulations are generally only a few hundred years long. However, modeling the long-term equilibration in response to radiative forcing perturbation is important for understanding many climate phenomena, such as the evolution of ocean circulation, time- and temperature-dependent feedbacks, and the differentiation of forced signal and internal variability. The aim of LongRunMIP is to facilitate research into these questions by serving as an archive for simulations that capture as much of this equilibration as possible. The only requirement to participate in LongRunMIP is to contribute a simulation with elevated, constant CO2 forcing that lasts at least 1,000 years. LongRunMIP is an MIP of opportunity in that the simulations were mostly performed prior to the conception of the archive without an agreed-upon set of experiments. For most models, the archive contains a preindustrial control simulation and simulations with an idealized (typically abrupt) CO2 forcing. We collect 2D surface and top-of-atmosphere fields and 3D ocean temperature and salinity fields. Here, we document the collection of simulations and discuss initial results, including the evolution of surface and deep ocean temperature and cloud radiative effects. As of October 2019, the collection includes 50 simulations of 15 models by 10 modeling centers. The data of LongRunMIP are publicly available. We encourage submissions of more simulations in the future.