The widespread availability of generative artificial intelligence (GenAI) has destabilized long-standing assumptions underlying assessment design in higher education, particularly the assumption that unsupervised written work reliably demonstrates individual student learning. This review examines how higher education institutions and scholars have responded to this challenge between 2023 and 2026, tracing a shift from initial detection-based responses toward structural assessment redesign. It synthesizes evidence on the effectiveness of two commonly proposed solutions, authentic assessment and process-oriented or oral assessment, finding that authentic assessment alone is not a reliable safeguard against undisclosed GenAI use, while process-based and interactive formats show more consistent, though not unlimited, resistance to substitution. The review also considers the instructional-design implications of treating GenAI as an unavoidable feature of the learning environment rather than an external threat to be excluded, and argues that effective assessment redesign is inseparable from broader questions about what GenAI is being asked to do within a given task. Implications for policy, faculty development, and future research are discussed.