The progress of the two-phase flow research has been divided into four stages in this series of the report. These periods are (I) 1948-1959, (II) 1960-1970, (III) 1971-1979, and (IV) 1980-1988. In this report, the development of analytical methods for flow instabilities in two-phase flow in periods (II),(III), and (IV) have been described. The development of the theoretical analysis in periods (II) and (III) are discussed based mainly on flow satability maps obtained by these analyses, and also the transition of the analytical methods are explained in relation to the rapid development of digital computer and the thermo-hydraulic simulation codes. In period (IV), the improvement of the modeling of fundamental equations and constitution equations and numerical calculation method as to the simulation codes is also described.
