Wine is widely consumed due to its distinctive sensory characteristics. However, during wine production and storage, several defects can appear. These can be the result of unwanted microbiological activity or due to the unbalanced levels of some compounds resulting from an unbalanced grape chemical composition or inadequate winemaking practices and storage conditions. The main purpose of wine stabilisation is the removal of wine defects, either visual, olfactive, gustative, or tactile, the increase in wine safety and stability by fining and filtration operations, avoiding the occurrence of some usual wine precipitations after bottling. Although the best strategy is to prevent the appearance of wine defects, when present, several fining agents or additives, and technologies are available today with different performances and impact on wine quality. By physicochemical and sensory analysis, the defect is detected, and if the objective is removing them, some laboratory trials are performed to achieve a better treatment approach. This review overviews the principal wine defects and treatments available today and in the near future. Generally, the future trend is the use of more sustainable and environmentally friendly fining agents and technologies, looking for treatments with better performance and specificity.