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Carbon nanotube field-effect transistors (CNFETs) are excellent candidates for building highly energy-efficient electronic systems of the future. Fundamental limitations inherent to carbon nanotubes (CNTs) pose major obstacles to the realization of robust CNFET digital very large-scale integration (VLSI): 1) it is nearly impossible to guarantee perfect alignment and positioning of all CNTs despite near-perfect CNT alignment achieved in recent years; 2) CNTs can be metallic or semiconducting depending on chirality; and 3) CNFET circuits can suffer from large performance variations, reduced yield, and increased susceptibility to noise. Today's CNT process improvements alone are inadequate to overcome these challenges. This paper presents an overview of: 1) imperfections and variations inherent to CNTs; 2) design and processing techniques, together with a probabilistic analysis framework, for robust CNFET digital VLSI circuits immune to inherent CNT imperfections and variations; and 3) recent experimental demonstration of CNFET digital circuits that are immune to CNT imperfections. Significant advances in design tools can enable robust and scalable CNFET circuits that overcome the challenges of the CNFET technology while retaining its energy-efficiency benefits.