By tradition, herbal infusions have been mainly consumed for their pleasant taste, but, nowadays, the consumer, along with the pleasantness of drinking a savory beverage, also looks for their health benefits. Grapes and grape/wine by-products are a rich source of health-promoting compounds, presenting great potential for the development of new beverages. Moreover, grape-infusion preparation is no more than a sustainable or green way of extracting polyphenols and other nutraceutical compounds from grapes and grape leaves. In this review, we summarize the benefits of drinking grape infusions and discuss the sustainable processes of extracting potential nutraceutical compounds from grapes and grape by-products, which are often considered fermentation waste and are discarded to the environment without proper treatment.