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Masonry construction, the assembly of discrete units into load-bearing structures, offers distinctive advantages for sustainable building: local material sourcing, adaptability to diverse resources, and reduced embodied carbon when using minimally processed materials. This review traces the evolution of robotic masonry from early bricklaying machines of the 1980s to emerging systems capable of assembling irregular natural stones. We identify three persistent challenges that have limited the adoption of bricklaying robots: sensing and feedback for error compensation, computational planning for design-to-fabrication translation, and economic viability given high setup costs. We then present robotic stone masonry construction as a sustainability-driven frontier, detailing recent advances in geometric planning, manipulation of irregular units, and multi-sensor feedback control that have achieved sub-4 cm placement tolerance. The review argues that the next phase of masonry robotics lies in extending automation beyond uniform manufactured units toward low-carbon, locally-sourced materials, a shift requiring integrated sensing, planning, and design capabilities.