Aerodynamic optimization of compressors and turbines is critical for enhancing overall aero-engine performance. While current mainstream optimization methods are limited to performance improvements via shape modification， topology optimization is unconstrained by shape limitations and can seek optimal aerodynamic layouts and geometries within the entire design domain. Two topology optimization methods are developed specifically for aero-engine blade design. First， a density-based topology optimization method for compressible turbulent flows is presented. Building upon conventional variable-density approaches， this method explicitly incorporates compressible and turbulent characteristics. With design variables distributed across grid points in the design space， a discrete adjoint method is employed to achieve fluid topology optimization applicable from subsonic to supersonic regimes. When applied to turbine blades and rotor tip clearance configurations， this method demonstrates significant improvements in turbine efficiency. Second， a shape-topology parameterization method is introduced， proposing a novel technical route based on the concept of “aerodynamic primitives plus topological factors.” Using blade profiles as basic units and parameterizing topological factors， this approach treats both shape and spatial topology parameters as design variables. This effectively circumvents challenges inherent to density-based methods， such as blurred fluid-solid interfaces and stability issues in solving adjoint equations. For compressor cascade optimization， this method successfully generates topological layouts that balance loss and diffusion.