In this paper, a technique to model strong and weak discontinuities with the scaled boundary finite element method through enrichment is proposed. The main advantage of the method is that the enriched elements, in the spirit of the extended finite element method (XFEM), do not need to physically conform to the geometry of features, e.g. internal interfaces and cracks, and remeshing is unnecessary as the interfaces evolve. All the advantages of the SBFEM and the XFEM are retained. The stress singularity at the crack tip can be captured accurately and the stress intensity factors (SIFs) can be directly computed based on the singular displacement or stress at the crack tip within the framework of the SBFEM. The numerical properties and performance for the proposed method are assessed using several numerical examples. In particular, problems with discontinuities, e.g. voids, inclusions, and cracks are analysed. The results show that the accuracy and convergence rate of the new approach for solving void or inclusion problems are identical to those of the XFEM, but requires less number of degrees-of-freedom than the XFEM. For crack problems, compared with the XFEM with topological enrichment, the developed method is superior.