The past two decades of 3D global simulations of convective dynamos have improved our understanding of how stars and planets generate their magnetic fields. However, no global dynamo simulation has yet been able to afford the spatial resolution required to simulate the turbulent convection which exists in the low-viscosity fluid interiors of these bodies. They have all employed greatly enhanced "eddy" diffusivities to crudely account for the transport and mixing by the small unresolved (subgrid scale) turbulence and to numerically stabilize the low resolution numerical solutions. Consequently convection in most numerical simulations has been laminar with spatial scales comparable to the size of the convection zone instead of being a broad spectrum of anisotropic heterogeneous turbulence. Many dynamo simulations have also ignored density stratification, which is a major source of vorticity in rotating turbulent convection. So, how robust are the results of the current models? We will not know until next-generation dynamo models are run at much higher spatial resolution and much lower viscosity, which will require more computational resources and improved numerical methods. In the mean time we can get some insight by testing 2D models, which, although lack the correct geometry, can simulate strong turbulence within a large density stratification. Current 2D tests reveal important issues regarding convective penetration, gravity waves, Alfven waves and the maintenance of differential rotation.