Coupled ocean-atmosphere general circulation models (coupled GCMs) with interactive sea ice are the primary tool for investigating possible future global warming and numerous other issues in climate science. A long-standing problem with such models is that when different components of the physical climate system are linked together, the simulated climate can drift away from observation unless constrained by ad hoc adjustments to interface fluxes. However, 11 modern coupled GCMs, including three that do not employ flux adjustments, behave much better in this respect than the older generation of models. Surface temperature trends in control run simulations (with external climate forcing such as solar brightness and atmospheric carbon dioxide held constant) are small compared with observed trends, which include 20th century climate change due to both anthropogenic and natural factors. Sea ice changes in the models are dominated by interannual variations. Deep ocean temperature and salinity trends are small enough for model control runs to extend over 1000 simulated years or more, but trends in some regions, most notably the Arctic, differ substantially among the models and may be problematic. Methods used to initialize coupled GCMs can mitigate climate drift but cannot eliminate it. Lengthy "spin-ups" of models, made possible by increasing computer power, are one reason for the improvements this paper documents.