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 [1] and under the limit of large Stefan number as treated first by Anderson and Worster [2]. As determined by Riahi [3] over an extensive range of the parameter values, the oscillatory mode of convection corresponds to smaller critical value R_c 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 [4] 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 [5] indicated preference of either supercritical simple travelling rolls or supercritical standing rolls depending on the particular parameter values of the problem.