Only a mere fraction of the huge variety of human pathogenic viruses can be targeted by the currently available spectrum of antiviral drugs. The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) outbreak has highlighted the urgent need for molecules that can be deployed quickly to treat novel, developing or re-emerging viral infections. Sulfated polysaccharides are found on the surfaces of both the susceptible host cells and the majority of human viruses, and thus can play an important role during viral infection. Such polysaccharides widely occurring in natural sources, specifically those converted into sulfated varieties, have already proved to possess a high level and sometimes also broad-spectrum antiviral activity. This antiviral potency can be determined through multifold molecular pathways, which in many cases have low profiles of cytotoxicity. Consequently, several new polysaccharide-derived drugs are currently being investigated in clinical settings. We reviewed the present status of research on sulfated polysaccharide-based antiviral agents, their structural characteristics, structure–activity relationships, and the potential of clinical application. Furthermore, the molecular mechanisms of sulfated polysaccharides involved in viral infection or in antiviral activity, respectively, are discussed, together with a focus on the emerging methodology contributing to polysaccharide-based drug development.