This work is devoted to the investigation and solution of differential games with a vector-valued pay-off function. Slater-, Pareto- and Geoffrion-optimal strategies are defined for such a game, and their positive and negative properties are examined. A new class of solutions called vector-valued guarantees is proposed and their properties examined. According to the authors, this new approach is superior primarily because it has the properties of equivalence and interchangeability - antagonistic players should be able to achieve guaranteed results simultaneously. The approach is tested in a competition problem, as well as a pursuit game with noise. This book should be useful for students, postgraduates and specialists who are investigating the different fields of applied mathematics, control theory, economics and near-by directions of scientific knowledge.