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Brittle fracture is the cracking of a metallic object or other elastic material under stress where the material exhibits little or no evidence of plastic deformation before the fracture occurs. It usually occurs rapidly and can be catastrophic in engineering practice. Understanding the initiation and propagation of brittle fracture is vital to the prevention of fracture failure in engineering design, however its mathematical modeling has been challenging due to the discontinuity nature of cracking. In recent years the phase-field mathematical modeling of brittle fracture has gained considerable attention, where a phase-field function is introduced to characterize the damage level of the material, cracks are represented as sharp but continuous layers, and the total free energy is modified to involve cracking effects. In this talk, I will discuss some of the mathematical and computational challenges with the phase-field modeling of brittle fracture. We will also present an adaptive moving mesh finite element method for the numerical solution of the model and numerical results obtained with the method.