Assessment and feedback activities in higher education are undergoing significant changes. Many universities and institutes still rely on traditional testing and grading methods, which often fall short in supporting meaningful student learning, especially in large classes. Although educational policies, such as those promoted by the Bologna process, encourage more feedback-oriented and student-centered approaches, these practices can be difficult to implement due to time constraints and limited resources. Generative Artificial Intelligence (GenAI), particularly Large Language Models (LLMs), has shown strong potential in addressing these challenges. This review examines 21 research studies published between 2023 and 2025 that explore the use of GenAI in providing feedback and assessing student work in higher education, with some studies also comparing GenAI's performance with human instructors. Findings show that LLMs can generate personalized and constructive feedback and/or assist with fair and consistent assessment. However, in most studies, teachers still play a key role, as expert oversight is essential to ensure that grading assessments and assignment feedback are accurate, relevant, and aligned with learning objectives. For GenAI to be used effectively, educators need to understand how to work with these tools, such as learning GenAI prompt design and the basic principles behind LLMs. We recommend that academic institutions provide training for educators in AI literacy, prompt engineering, and the development of teaching strategies that combine the strengths of human judgment with AI support. By effectively integrating LLM tools, major assessment challenges, such as limited time and inconsistent feedback quality, can be addressed while also enhancing student learning and engagement.