Multiple sclerosis (MS) is a chronic, inflammatory, demyelinating disease of autoimmune origin, in which inflammation and demyelination lead to neurodegeneration and progressive disability. Treatment is aimed at slowing down the course of the disease and mitigating its symptoms. One of the first-line treatments used in patients with MS is glatiramer acetate (GA). However, in clinical practice, a response rate of between 30% and 55% is observed. This variability in the effectiveness of the medication may be influenced by genetic factors such as polymorphisms in the genes involved in the pathogenesis of MS. Therefore, this review assesses the impact of genetic variants on the response to GA therapy in patients diagnosed with MS. The results suggest that a relationship exists between the effectiveness of the treatment with GA and the presence of polymorphisms in the following genes: CD86, CLEC16A, CTSS, EOMES, MBP, FAS, TRBC1, IL1R1, IL12RB2, IL22RA2, PTPRT, PVT1, ALOX5AP, MAGI2, ZAK, RFPL3, UVRAG, SLC1A4, and HLA-DRB1*1501. Consequently, the identification of polymorphisms in these genes can be used in the future as a predictive marker of the response to GA treatment in patients diagnosed with MS. Nevertheless, there is a lack of evidence for this and more validation studies need to be conducted to apply this information to clinical practice.