This work investigates the existing capabilities and limitations in numerical modeling of fracture problems in functionally graded materials (FGMs) by means of the well-known finite element code ABAQUS. Quasi-static crack initiation and growth in planar FGMs is evaluated. Computational results of fracture parameters are compared to experimental results and good agreement is obtained. The importance of the numerical fit of the elastic properties in the FE model is analyzed in depth by means of a sensitivity study and a novel method is presented. Several key computational issues derived from the continuous change of the material properties are also addressed and the source code of a user subroutine USDFLD is provided in the Appendix for an effective implementation of the property variation. The crack propagation path is calculated through the extended finite element method and subsequently compared to available experimental data. Suitability of local fracture criteria to simulate crack trajectories in FGMs is discussed and a new crack propagation criterion is suggested.