Abstract: Most stars and planets exhibit some degree of magnetism. In all probability, these magnetic fields are generated by hydromagnetic dynamo action, whereby the inductive effects due to the motions of electrically-conducting fluids in the astrophysical body are able to sustain magnetic fields against the action of ohmic dissipation. The surprising aspect of this is that many astrophysical dynamos seem to produce large-scale ordered magnetic fields rather than simply small-scale disordered fields that are structured on the scale of the underlying flow. Motivated by these considerations, I will describe a prototypical dynamo problem, namely dynamo action driven by convection in an electrically-conducting fluid. I will talk about the circumstances under which such a system can drive small- and large-scale dynamos, comparing results from simulations with observations of solar and planetary magnetic fields. I will also relate these results to standard mean-field dynamo theory, describing some of the limitations that are associated with this approach.