This paper is showing the results of a study focused on the occurrence of failures in heterogeneous structural materials, especially on the issue of modelling the formation and propagation of cracks. Attention is paid to the application of the finite element method to various types of materials in order to find critical parameters determining the behaviour of materials during the damage process. Application of damage mechanisms and approaches to modelling the origin of crack propagation by modifications in FEM systems are presented. Furthermore, some practical applications are tested. The main effort is devoted to composites and searching for procedures when using cohesion models for their more accurate modelling, especially near the stress concentrators, or before the crack tip. The use of the cohesive approach of elements that represent the physical nature of energy release in front of the crack front has proven to be promising not only in the direct use of cohesive elements, but also in combination with modified methods of standard finite elements. The authors of this article use their long-term experience in the given issue, which they consistently confront with developments in this area in recent years.