(1) Introduction: Wilson’s disease (WND) is an autosomal recessive disorder of copper (Cu) metabolism. Many tools are available to diagnose and monitor the clinical course of WND. Laboratory tests to determine disorders of Cu metabolism are of significant diagnostic importance. (2) Methods: A systematic review of the literature in the PubMed, Science Direct, and Wiley Online Library databases was conducted. (Results): For many years, Cu metabolism in WND was assessed with serum ceruloplasmin (CP) concentration, radioactive Cu test, total serum Cu concentration, urinary copper excretion, and Cu content in the liver. The results of these studies are not always unambiguous and easy to interpret. New methods have been developed to calculate non-CP Cu (NCC) directly. New parameters, such as relative Cu exchange (REC), reflecting the ratio of CuEXC to total serum Cu, as well as relative Cu exchange (REC), reflecting the ratio of CuEXC to total serum Cu, have been shown to be an accurate tool for the diagnosis of WND. Recently, a direct and fast LC-ICP-MS method for the study of CuEXC was presented. A new method to assess Cu metabolism during treatment with ALXN1840 (bis-choline tetrathiomolybdate [TTM]) has been developed. The assay enables the bioanalysis of CP and different types of Cu, including CP-Cu, direct NCC (dNCC), and labile bound copper (LBC) in human plasma. Conclusions: A few diagnostic and monitoring tools are available for patients with WND. While many patients are diagnosed and adequately assessed with currently available methods, diagnosis and monitoring is a real challenge in a group of patients who are stuck with borderline results, ambiguous genetic findings, and unclear clinical phenotypes. Technological progress and the characterization of new diagnostic parameters, including those related to Cu metabolism, may provide confidence in the more accurate diagnosis of WND in the future.