The path planning with homotopy class constraints is an important technical means for the path planning for space robots. It can effectively achieve lightweight computation of path planning under complex constraints， and has important theoretical significance and application value. In the path planning with homotopy class constraints， the set of robot's feasible paths is limited within a homotopy structure with good functional properties. The search for optimized paths is carried out within the homotopy structure， which ensures the effectiveness of the path planning while reducing the complexity of the algorithm. In this paper， the approximation algorithm frameworks for homotopy-based path planning， the path homotopy construction methods， and the solutions to the planning problems are introduced. The progress of homotopy path planning for space robots in recent years is analyzed from two aspects， i.e.， path homotopy construction methods with constrains and homotopy optimization search. According to the current research situation， the future development of homotopy path planning methods for space robots is prospected.