Food packaging is a multidisciplinary area that encompasses food science and engineering, microbiology, as well as chemistry, and ignited tremendous interest in maintaining the freshness and quality of foods and their raw materials from oxidation and microbial spoilage. With the advances in the packaging industry, they could be engineered as easy-to-open, resealable, active, as well as intelligent with the incorporation of sensory elements while offering desired barrier properties against oxygen and water vapor. In this regard, the use of the electrospinning approach allows producing nanofibrous packaging materials with large surface-to-volume ratios and enables the higher loading of active agents into packaging materials. Electrospun packaging materials have been produced from various polymers (i.e., synthetic and natural) and their (nano)composites, and were mainly exploited for the encapsulation of active agents for their use as active food packaging materials. The electrospinning process was also used for the deposition of electrospun fibers on films to enhance their performance (e.g., as reinforcement material, or to enhance barrier properties). They could be even engineered to provide nutraceuticals to food, or antioxidant, antimicrobial or antifungal protection to the packaged food. In this article, first, introductory descriptions of food packaging, barrier properties, and electrospinning are given. Afterward, active and intelligent food packaging materials are briefly discussed, and the use of electrospinning for the fabrication of active food packaging materials is elaborated. Particular interest has been given to the polymer-type used in the production of electrospun fibers and active properties of the resultant packaging materials (e.g., antioxidant, antibacterial, antifungal). Finally, this review paper concludes with a summary and future outlook towards the development of electrospun food packaging materials.