Submesoscale processes have been extensively studied in observations and simulations of fronts. Recent idealized simulations show that submesoscale instabilities also occur in baroclinic mesoscale cyclones and anticyclones. The instabilities in the anticyclone grow faster and at coarser grid resolution than in the cyclone. The instabilities lead to larger restratification in the anticyclone than in the cyclone. The instabilities also lead to changes in the mean azimuthal jet around the anticyclone from 2-km resolution, but a similar effect only occurs in the cyclone at 0.25-km resolution. A numerical passive tracer experiment shows that submesoscale instabilities lead to deeper subduction in the interior of anticyclonic than cyclonic eddies because of outcropping isopycnals extending deeper into the thermocline in anticyclones. An energetic analysis suggests that both vertical shear production and vertical buoyancy fluxes are important in anticyclones but primarily vertical buoyancy fluxes occur in cyclones at these resolutions. The energy sources and sinks vary azimuthally around the eddies caused by the asymmetric effects of the Ekman buoyancy flux. Glider transects of a mesoscale anticyclone in the Tasman Sea show that water with low stratification and high oxygen concentrations is found in an anticyclone, in a manner that may be consistent with the model predictions for submesoscale subduction in mesoscale eddies.