Although melanoma is one of the most immunogenic tumors, it has an ability to evade anti-tumor immune responses by exploiting tolerance mechanisms, including negative immune checkpoint molecules. The most extensively studied checkpoints represent cytotoxic T lymphocyte-associated protein-4 (CTLA-4) and programmed cell death protein 1 (PD-1). Immune checkpoint inhibitors (ICI), which were broadly applied for melanoma treatment in the past decade, can unleash anti-tumor immune responses and result in melanoma regression. Patients responding to the ICI treatment showed long-lasting remission or disease control status. However, a large group of patients failed to respond to this therapy, indicating the development of resistance mechanisms. Among them are intrinsic tumor properties, the dysfunction of effector cells, and the generation of immunosuppressive tumor microenvironment (TME). This review discusses achievements of ICI treatment in melanoma, reasons for its failure, and promising approaches for overcoming the resistance. These methods include combinations of different ICI with each other, strategies for neutralizing the immunosuppressive TME and combining ICI with other anti-cancer therapies such as radiation, oncolytic viral, or targeted therapy. New therapeutic approaches targeting other immune checkpoint molecules are also discussed.