In this contribution, we discuss the modeling of discontinuities using non-conforming meshes. For weak discontinuities we propose to regularize the sharp material interface over a characteristic length scale. To define the material properties in the transition region, homogenization assumptions are used that fulfill the kinematic compatibility across the interface and the static equilibrium at the interface. For strong discontinuities we use a phase-field model that is able to account for failure of adhesive interfaces and extend it to general heterogeneous materials. To improve the efficiency of the approach, adaptive isogeometric analysis is used to discretize the computational domain.