(1) Introduction: Wilson’s disease (WND) is an autosomal recessive disorder of copper metabolism. The WND gene is ATP7B, located on chromosome 13. WND is characterized by high clinical variability, which causes diagnostic difficulties. (2) Methods: The PubMed, Science Direct, and Wiley Online Library medical databases were reviewed using the following phrases: “Wilson’s disease”, “ATP7B genotype”, “genotype-phenotype”, “epigenetics”, “genetic modifiers”, and their combinations. Publications presenting the results of experimental and clinical studies, as well as review papers, were selected, which concerned: (i) the diversity of genetic strategies and tests used in WND diagnosis; (ii) the difficulties of genetic diagnosis, including uncertainty as to the pathogenicity of variants; (iii) genetic counseling; (iv) phenotypic effects of ATP7B variants in patients with WND and in heterozygous carriers (HzcWND); (v) genetic and epigenetics factors modifying the clinical picture of the disease. (3) Results and conclusions: The genetic diagnosis of WND is carried out using a variety of strategies and tests. Due to the large number of known variants in the ATP7B gene (>900), the usefulness of genetic tests in routine diagnostics is still relatively small and even analyses performed using the most advanced technologies, including next-generation sequencing, require additional tests, including biochemical evidence of abnormal copper metabolism, to confirm the diagnosis of WND. Pseudodominant inheritance, the presence of three various pathogenic variants in the same patient, genotypes indicating the possibility of segmental uniparental disomy, have been reported. Genotype–phenotype relationships in WND are complex. The ATP7B genotype, to some extent, determines the clinical picture of the disease, but other genetic and epigenetic modifiers are also relevant.