Cohesive fracture, characterized by opening-dependent cohesive tractions at an extended crack tip, is a significant phenomenon in metals and plastic materials, leading to the formation of complex multiscale structures. While brittle fracture has long been a classical subject in applied mathematics, the study of cohesive fracture models has only recently gained attention. This project will explore novel phase-field models of cohesive fracture, incorporating mechanically motivated, highly non-linear damage functions, and examine their sharp-interface limits. The research will span across modeling, analysis, and simulation.