Peridynamics is a nonlocal continuum mechanics formulation known for its ability to model damage and fracture phenomena. However, many peridynamic formulations suffer from significant discretization errors in elastic modeling due to nonlocality and surface effects. Several approaches exist to mitigate these limitations, such as the correspondence formulation, which utilizes an approximated deformation gradient to calculate stress forces. Although this reformulation improves the accuracy of elastic modeling, it introduces instabilities. The bond-associated peridynamic formulation is a promising solution, as it combines the advantages of bond-based modeling for fracture with the accuracy of the correspondence formulation for elastic modeling without instabilities. In standard peridynamics formulations, crack propagation is mostly modeled with a critical bond strain criterion. With bond-associated modeling, this can be reconsidered, as stress tensors are calculated for each bond, making it possible to introduce new criteria. This study presents a comprehensive review of crack propagation criteria, especially for bond-associated peridynamics. Explanatory examples and interesting simulations are provided, showcasing the capabilities and practical relevance.