Evolving discontinuities can be modelled in a truly discrete sense, or in a smeared or continuum manner. Within the class of discrete models, cohesive-surface approaches are very versatile, in particular for heterogeneous materials. However, limitations exist, in particular related to stress triaxiality, which cannot be captured well in standard cohesive-surface models. We will therefore discuss an elegant enhancement of the cohesive-surface model to include stress triaxiality, which preserves the discrete character of cohesive-surface models. Subsequently, we will outline how the cohesive approach to fracture can be extended to multi-phase media, in particular fluid-saturated porous media. Whether a discontinuity is modelled via a continuum model, or in a discrete manner, advanced discretisation methods are needed to model the internal free boundary. Level sets, extended finite element methods and isogeometric analysis are important and promising tools in this respect. Examples will be given, including delamination in layered shells and fracture in fluid-saturated media. In smeared approaches to fracture higher-order spatial gradients typically evolve. Here, isogeometric analysis offers advantages by virtue of the smoothness of its basis functions, as will be demonstrated at the hand of a gradient-enhanced continuum damage model. In addition to approaches like NURBS that exploit tensor products for multi-dimensional generalisations, Powell-Sabin B-splines seem to be versatile, since they are defined on triangles, and thus share the ease and flexibility of mesh generation that characterises standard triangular finite elements. Another development in continuum approaches to fracture is the phase-field theory. We will discuss the formulation for brittle fracture, including its relation to gradient damage models, and extend the phase-field approach to cohesive fracture-surface models. We conclude with a concise discussion of the advantages of phase-field theories for damage and fracture
