<p>The exploration of ice giant systems represents one of the priority areas in deep space exploration for the coming decade. Owing to the vast orbital distances of these planets and the need for extensive in-system transfers during missions, trajectory design and optimization constitute a critical enabling technology for the exploration of ice giant systems. While numerous mission concepts and orbital design methodologies have been proposed to date, a comprehensive review of these methodologies is currently lacking, which hinders the further development of novel design techniques and the formulation of new mission proposals. This survey systematically synthesizes both established and state-of-the-art methods across four primary mission phases: interplanetary transfer, ice giant capture, planetary satellite tours, and other scientific observations targeting planets and comets. For each phase, different design strategies are introduced, with their advantages and capabilities described and analyzed to reveal technological progress. Finally, perspectives on future developments are provided, aiming to establish a reference framework for further research in trajectory design and optimization for ice giant system exploration.</p>