Steady and oscillatory modes of convective flow in a horizontal mushy layer during alloy solidification are reviewed. The theoretical problem has been based on the model developed by Amberg and Homsy and under the limit of large Stefan number as treated first by Anderson and Worster. As determined by Riahi over an extensive range of the parameter values, the oscillatory mode of convection corresponds to smaller critical value Rc of the Rayleigh number at the onset of motion as compared to the corresponding one for the steady mode of convection. Nonlinear steady convection studied by Anderson and Worster indicated preference of subcritical steady hexagon pattern convection for sufficiently small magnitude |e| of the amplitude of convection, while supercritical steady two-dimensional rolls were found to be stable for |e| beyond some value. Nonlinear oscillatory convection studied by Riahi indicated preference of either supercritical simple travelling rolls or supercritical standing rolls depending on the particular parameter values of the problem.