Current antiviral therapy research is focused on developing dosage forms that enable highly effective drug delivery, providing a selective effect in the organism, lower risk of adverse effects, a lower dose of active pharmaceutical ingredients, and minimal toxicity. In this article, antiviral drugs and the mechanisms of their action are summarized at the beginning as a prerequisite background to develop relevant drug delivery/carrier systems for them, classified and briefly discussed subsequently. Many of the recent studies aim at different types of synthetic, semisynthetic, and natural polymers serving as a favorable matrix for the antiviral drug carrier. Besides a wider view of different antiviral delivery systems, this review focuses on advances in antiviral drug delivery systems based on chitosan (CS) and derivatized CS carriers. CS and its derivatives are evaluated concerning methods of their preparation, their basic characteristics and properties, approaches to the incorporation of an antiviral drug in the CS polymer as well as CS nanoparticulate systems, and their recent biomedical applications in the context of actual antiviral therapy. The degree of development (i.e., research study, in vitro/ex vivo/in vivo preclinical testing), as well as benefits and limitations of CS polymer and CS nanoparticulate drug delivery systems, are reported for particular viral diseases and corresponding antivirotics.