Abstract Generalized (viscosity) solutions of Hamilton-Jacobi and Bellman-Isaacs equations are defined by means of pairs of differential inequalities. This type of solution exists, is unique and in the case of the Bellman-Isaacs equation, is identical with the value function of the corresponding differential game. Unlike the published literature (/1–8/ etc.), in which continuous solutions were considered, differential games are considered here with semicontinuous payoff functions and correspondingly semicontinuous solutions. Definitions are introduced for these solutions and existence and uniqueness theorems are proved. Problems in which the value function (Bellman function) may fail to be continuous are well-known. For example, in a differential game of pursuit-evasion the payoff function, defined as the time to capture, is lower semicontinuous and the corresponding value function is also lower semicontinuous. In particular, the Bellman function of the optimum response problem is lower semicontinuous; its properties have been studied by many authors, and differential relations have been derived /9, 10/ that represent the optimality principle in the response problem. The theory of differential games provided a suitable framework for studying many questions in the general theory of Hamilton-Jacobi equations. The aim of this paper is to develop an apparatus of differential inequalities and define generalized (viscosity) solutions for the case in which these solutions are semicontinuous. To fix our ideas, a differential game with fixed terminal time is considered. Results are then formulated for the game of pursuit-evasion.