Magnetoconvection, the study of the interaction between thermal convection and magnetic fields, was originally motivated by the dynamics of magnetic fields in the solar photosphere. Figure 1 (see color insert following page 234) shows a recent image from the Swedish Solar Telescope (courtesy of T.E. Berger) showing the solar granulation, sunspots (dark) and bright points in downdrafts between supergranular and intergranular lanes. The last two of these are signatures of magnetic field structures. The challenge to theorists and numericists is to understand these structures, to relate them to observations of emission spectra etc., to use them as diagnostics for behavior deeper down in the solar convection zone, and ultimately to relate the phenomena to the solar dynamo responsible for the sunspot cycle. This last is beyond the scope of the present review, but I have tried to survey in some breadth work done in the last twenty years on the effects of an imposed field. In fact, interest in magnetoconvection as a bifurcation problem, similar to double-diffusive convection, has led to a large number of papers that are not directly astrophysical, and I have tried to cover these as well, and crave the reader’s indulgence for emphasizing work with which I have been involved. For more details, the reviews by Proctor and Weiss (1982), Hughes and Proctor (1988), Weiss (1997) and Schu¨ssler and Kno¨lker (2001) may be consulted.