In this review, we summarize recent progress in crystal growth and understanding of the influence of crystal structure on superconductivity in pure and Pb-doped Bi₂Sr₂CuO_y (Bi-2201) materials belonging to the overdoped region of high-temperature cuprate superconductors. The crystal growth of Bi-2201 superconductors faces challenges due to intricate materials chemistry and the lack of knowledge of corresponding phase diagrams. Historically, a crucible-free floating zone method emerged as the most promising growth approach for these materials, resulting in high-quality single crystals. This review outlines the described methods in the literature and the authors’ synthesis endeavors encompassing Pb-doped Bi-2201 crystals, provides a detailed structural characterization of as-grown and post-growth annealed samples, and highlights optimal growth conditions that yield large-size, single-phase, and compositionally homogeneous Bi-2201 single crystals.