Autophagy is a critical cellular mechanism that regulates the degradation of misfolded and aggregated proteins and non-functional intracellular organelles. Based on the fundamental qualities of the substrates targeted for degradation and the distinct molecular mechanisms involved, autophagy can be classified into three major types: macroautophagy, microautophagy, and chaperone-mediated autophagy (CMA). Sequestosome 1 (SQSTM1)/p62, which functions as a signaling hub integrating nuclear factor kappa B (NF-κB), the mechanistic target of rapamycin complex 1 (mTORC1), and Kelch-like ECH-associated protein 1 (Keap1)–nuclear factor erythroid 2–related factor 2 (NRF2) pathways, serves as a selective macroautophagy/autophagy receptor that binds ubiquitinated cargo proteins and recruits them to the autophagosome for subsequent degradation in the autolysosome. Furthermore, the phase separation of p62 is an important regulatory process in the autophagy mechanism, but recent studies have demonstrated that impaired or excessive autophagy mediated by p62 is associated with cancer development. This review summarizes the role of autophagy—including its types, mechanisms, and the pathway related to the ubiquitin-dependent selective autophagy receptor p62—in cancer progression.