This review paper discusses the basic problems related to the use of cohesive models to simulate the initiation and development of failure in various types of engineering issues. The most commonly used cohesive zone models (CZMs) are described. Recent achievements in the field of cohesive modeling are characterized, with particular emphasis on the problem of mixed mode loading, the influence of the strain rate, the stress state triaxiality, and fatigue. A separate chapter of the work is devoted to the identification of cohesive parameters. Examples of the use of CZMs for the analysis of the fracture and failure process in various applications, both on the macro and microscopic scale, are given. The directions of CZMs development were indicated as well as the issues that are currently under particularly intensive development.