This paper presents the lattice element models, as a class of discrete models, in which the structural solid is represented as an assembly of one-dimensional elements. This idea allows one to provide robust models for propagation of discontinuities, multiple cracks interaction or cracks coalescence. Many procedures for computation of lattice element parameters for representing linear elastic continuum have been developed, with the most often used ones discussed herein. Special attention is dedicated to presenting the ability of this kind of models to consider material disorder, heterogeneities and multi-phase materials, which makes lattice models attractive for meso- or micro-scale simulations of failure phenomena in quasi-brittle materials, such as concrete or rocks. Common difficulties encountered in material failure and a way of dealing with them in the lattice models framework are explained in detail. Namely, the size of the localized fracture process zone around the propagating crack plays a key role in failure mechanism, which is observed in various models of linear elastic fracture mechanics, multi-scale theories, homogenization techniques, finite element models, molecular dynamics. An efficient way of dealing with this kind of phenomena is by introducing the embedded strong discontinuity into lattice elements, resulting with mesh-independent computations of failure response. Moreover, mechanical lattice can be coupled with mass transfer problems, such as moisture, heat or chloride ions transfer which affect the material durability. Any close interaction with a fluid can lead to additional time dependent degradation. For illustration, the lattice approach to porous media coupling is given here as well. Thus, the lattice element models can serve for efficient simulations of material failure mechanisms, even when considering multi-physics coupling. The main peculiarities of such an approach have been presented and discussed in this work.