In the directional sodilification of binary alloys, several double-diffusive convective flows of various stability characteristics occur either in sequence or simultaneously. Each convective flow can seriously affect the crystalline of the resultant casting, and under some circumstances may in turn deteriorate the mechanical properties of the product. Similar convective flows are also found in, for example, the Earth inner core, the magma chamber, and the polar ocean, and so on. In this review, we discuss the characteristics of these convective flows from both experimental and theoretical viewpoints. A series of experimental results are first introduced to illustrate the flow structure as well as the convection phenomena. An extensive discussion on the theoretical model is then provided. Special attention is paid to the stability characteristics of each convective flow, which may be of fundamental interest to the fluid mechanics researchers. The interaction between the convection and the solidification is given to gain the insights into the thermodynamic aspect of the problem. Special cases based on the effects due to some physical parameters are provided. Finally, the difficulties encountered by previous investigations are discussed and some potential problems merit for future research are proposed.