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Material degradation is vital to many industries and computational modeling and simulation of it continues to be a challenge and active area of research. Despite some progress in simulating fracture and damage using classical models, realistic prediction of complex damage progression and failure has been out of reach for many decades. Peridynamics (PD), a nonlocal theory introduced in 2000, opened up new avenues in modeling material degradation and failure. Existing numerical methods used to discretize PD equations, however, are quite expensive as the PD nonlocal interactions make them unaffordable for large-scale 3D simulations. In this work, we first present a peridynamic (PD) analysis of interface fracture and determine key elements that are essential in reproducing the observed dynamic fracture behavior in bonded PMMA samples. For the first time, we show that the meshfree discretizations for corresponding peridynamic (PD) models lead to graph structures. A semi-analytical approach, similar to the one used in Spectral Graph (SG) approaches, is used to solve problems in transient heat diffusion. We next introduce a fast convolution-based method (FCBM) for peridynamic (PD) models of pitting corrosion damage. Similar simulations of actual, detailed, evolution of pittingcorrosion damage (not using rough, approximating 1D theories) can now be conducted at centimeter and meter scales, for the first time. We introduce PeriFast/Corrosion, an open source MATLAB code that uses the FCBM for peridynamic (PD) models of corrosion damage (uniform and pitting corrosion). For the first time, the FCBM is used for simulating corrosion in heterogeneous materials. To protect metals from corrosion, electrodeposition can be used to deposit a thin metal film on the metal substrate. We introduce a new peridynamic (PD) model of electrodeposition and solve it using the fast convolution-based method (FCBM).