This paper discusses material interpolation schemes in topology optimization, focusing on their relationship with variational bounds on effective properties of composite materials. The authors analyze and compare various approaches to grey-scale density-like interpolation functions used in topology optimization. They derive necessary conditions for the realization of grey-scale via composites, leading to a physical interpretation of feasible designs and optimal solutions. The paper shows that the so-called artificial interpolation model often falls within the framework of microstructurally based models. The paper discusses single and multi-material structural design in elasticity and multi-physics problems. It highlights the challenges of black-and-white designs, where the optimal topology consists of a macroscopic variation of one material and void. These designs are ill-posed due to the possibility of nonconvergent minimizing sequences. To address this, the solution space is restricted by imposing constraints on the complexity of admissible designs, such as limiting the perimeter or introducing filtering functions. The paper also discusses the use of relaxation techniques to achieve solution existence, allowing for microstructures and homogenized properties. These methods lead to designs that can only be realized with microstructure, but no definite length scale is associated with it. Relaxation provides a basis for direct synthesis where composite materials can be part of the final design. The paper compares various interpolation schemes with micromechanical models, showing that the commonly used 'fictitious material model' is misleading. It discusses single and multi-material topology design in elasticity and multi-physics problems, emphasizing the importance of understanding the physical relevance of interpolation schemes. The paper concludes that while computational schemes often produce designs with 'grey' regions, the physical realization of feasible designs is important when interpreting results from premature termination of optimization algorithms.This paper discusses material interpolation schemes in topology optimization, focusing on their relationship with variational bounds on effective properties of composite materials. The authors analyze and compare various approaches to grey-scale density-like interpolation functions used in topology optimization. They derive necessary conditions for the realization of grey-scale via composites, leading to a physical interpretation of feasible designs and optimal solutions. The paper shows that the so-called artificial interpolation model often falls within the framework of microstructurally based models. The paper discusses single and multi-material structural design in elasticity and multi-physics problems. It highlights the challenges of black-and-white designs, where the optimal topology consists of a macroscopic variation of one material and void. These designs are ill-posed due to the possibility of nonconvergent minimizing sequences. To address this, the solution space is restricted by imposing constraints on the complexity of admissible designs, such as limiting the perimeter or introducing filtering functions. The paper also discusses the use of relaxation techniques to achieve solution existence, allowing for microstructures and homogenized properties. These methods lead to designs that can only be realized with microstructure, but no definite length scale is associated with it. Relaxation provides a basis for direct synthesis where composite materials can be part of the final design. The paper compares various interpolation schemes with micromechanical models, showing that the commonly used 'fictitious material model' is misleading. It discusses single and multi-material topology design in elasticity and multi-physics problems, emphasizing the importance of understanding the physical relevance of interpolation schemes. The paper concludes that while computational schemes often produce designs with 'grey' regions, the physical realization of feasible designs is important when interpreting results from premature termination of optimization algorithms.