Characterizing mechanisms of protein homeostasis, a process of balancing between protein synthesis and protein degradation, is important for understanding the potential causes of human diseases. The ubiquitin–proteasome system (UPS) is a well-studied mechanism of protein catabolism, which is responsible for eliminating misfolded, damaged, or aging proteins, thereby maintaining quality and quantity of cellular proteins. The UPS is composed of multiple components, including a series of enzymes (E1, E2, E3, and deubiquitinase [DUB]) and 26S proteasome (19S regulatory particles + 20S core particle). An impaired UPS pathway is involved in multiple diseases, including cancer. Several proteasome inhibitors, such as bortezomib, carfilzomib, and ixazomib, are approved to treat patients with certain cancers. However, their applications are limited by side effects, drug resistance, and drug–drug interactions observed in their clinical processes. To overcome these shortcomings, alternative UPS inhibitors have been searched for in many fields. Copper complexes (e.g., CuET, CuHQ, CuCQ, CuPDTC, CuPT, and CuHK) are found to be able to inhibit a core component of the UPS machinery, such as 20S proteasome, 19S DUBs, and NPLOC4/NPL4 complex, and are proposed to be one class of metal-based anticancer drugs. In this review, we will summarize functions and applications of copper complexes in a concise perspective, with a focus on connections between the UPS and cancer.