The development of antimicrobial food packaging is a very important and current goal, but it still difficult to implement in practice. Reducing microbial contamination and preserving food quality are very important tasks for food manufacturers as the use of antimicrobial packaging can preserve the health of consumers. On the other hand, the difficulty of degrading packaging materials, leading to environmental pollution, is also an important problem. These problems can be solved by using biodegradable biopolymers and antimicrobial agents in the production of food packaging. Very suitable antimicrobial agents are grape seed and skin extracts as they have high antioxidant and antimicrobial capacity and are obtained from grape pomace, a waste product of winemaking. The present review presents the valuable bioactive compounds contained in grape seeds and skins, the methods used to obtain the extracts, and their antimicrobial and antioxidant properties. Then, the application of grape seed and skin extracts for the production of antimicrobial packaging is reviewed. Emphasis is placed on antimicrobial packaging based on various biopolymers. Special attention is also paid to the application of the extract of grape skins to obtain intelligent indicator packages for the continuous monitoring of the freshness and quality of foods. The focus is mainly placed on the antimicrobial properties of the packaging against different types of microorganisms and their applications for food packaging. The presented data prove the good potential of grape seed and skin extracts to be used as active agents in the preparation of antimicrobial food packaging.