The evolution of stars and planets is influenced by turbulent flows occurring in their interiors; these flows transport heat and are responsible for generating the magnetic fields observed in the stars and planets. Often, the interior flows are convective (e.g., all stars transport some of their energy by convection, as do gaseous giant planets) and convection is believed to be responsible for the driving of large-scale flows, as well as contributing to the dynamo process that sustains the observed magnetic fields. Moreover, stellar (and some planetary) interiors are highly stratified; the fluid density drops off over many scale heights which can have profound effects on the internal dynamics. Furthermore, stars and planets rotate (some very rapidly) which further influences the flow dynamics, heat transport and magnetic field generation. A number of observations provide interesting constraints and suggest that our current understanding of the interplay between rotation, magnetism, stratification and convection is not yet complete. In this talk I will describe some of the open questions in this area and our efforts to understand them using idealised mathematical models.