Copper (Cu) is a vital element required for cellular growth and development; however, even slight changes in its homeostasis might lead to severe toxicity and deleterious medical conditions. Cancer patients are typically associated with higher Cu content in serum and tumor tissues, indicating increased demand of cancer cells for this micronutrient. Cu is known to readily cycle between the +1 and +2 oxidation state in biological systems. The mechanism of action of Cu complexes is typically based on their redox activity and induction of reactive oxygen species (ROS), leading to deadly oxidative stress. However, there are a number of other biomolecular mechanisms beyond ROS generation that contribute to the activity of anticancer Cu drug candidates. In this review, we discuss how interfering with intracellular Cu balance via either diet modification or addition of inorganic Cu supplements or Cu-modulating compounds affects tumor development, progression, and sensitivity to treatment modalities. We aim to provide the rationale for the use of Cu-depleting and Cu-overloading conditions to generate the best possible patient outcome with minimal toxicity. We also discuss the advantages of the use of pre-formed Cu complexes, such as Cu-(bis)thiosemicarbazones or Cu-N-heterocyclic thiosemicarbazones, in comparison with the in situ formed Cu complexes with metal-binding ligands. In this review, we summarize available clinical and mechanistic data on clinically relevant anticancer drug candidates, including Cu supplements, Cu chelators, Cu ionophores, and Cu complexes.