<p>Distributed fixed-time and prescribed-time optimization has become a key focus in the study of multi-agent systems (MASs), enabling efficient and scalable solutions to optimization problems with guaranteed convergence within fixed-time and prescribed-time frames, respectively. This survey presents a thorough overview of distributed fixed-time and prescribed-time optimization methodologies, focusing on two key paradigms: time-invariant and time-varying cases. Specifically, the survey begins by exploring fundamental principles of fixed-time optimization of MASs, emphasizing their advantages over asymptotic and finite-time methods, particularly in scenarios with strict convergence-time requirements. Then, recent advances are presented in distributed fixed-time optimization, including second-order MASs, event-triggered control, and application to smart grids. Following that, representative results on distributed prescribed-time optimization are provided, which extend the fixed-time counterpart to a problem with user-defined settling time. Finally, some open challenges in the field, such as handling communication delays, cyber-physical threats, and nonconvexities, are identified that deserve further investigation.</p>