<p>In practical engineering, a variety of random uncertainties are frequently encountered. To automatically design a structure with desired performance that can deal with these uncertain disturbances, robust topology optimization (RTO) of continuum structures inevitably involves numerous random variables. However, conventional uncertainty quantification methods often result in considerable computational cost or inaccurate results when tackling RTO problems with high-dimensional random variables. To this end, this paper proposes a new efficient framework based on direct probability integral method (DPIM) and sequential approximate integer programming with trust region (SAIP-TR) method for addressing RTO design problems of continuum structures with high-dimensional random variables. Firstly, the expansion optimal linear estimation method is adopted to characterize spatially varying random field. Then, DPIM-based framework is devised to attack the challenging issue of statistical moment estimation with high-dimensional random variables. Finally, the SAIP-TR method is utilized for discrete variable topology optimization to obtain clear topology configurations that are easy to process and manufacture. Four typical examples considering high-dimensional load uncertainty, material uncertainty, and mixed uncertainties are illustrated. The results demonstrate that the proposed framework can efficiently and accurately calculate statistical moments and solve robust discrete variable topology optimization problems with 20 random variables and generate clear black-and-white designs. Its computational efficiency is significantly enhanced compared to the widely used polynomial chaos expansion method. In addition, this paper also reveals that the number of truncation terms in random field has a remarkable impact on the optimal topology design configurations of structures.</p>