Drug innovation traditionally follows a de novo approach with new molecules through a complex preclinical and clinical pathway. In addition to this strategy, drug repositioning has also become an important complementary approach, which can be shorter, cheaper, and less risky. This review provides an overview of drug innovation in both human and veterinary medicine, with a focus on drug repositioning. The evolution of drug repositioning and the effectiveness of this approach are presented, including the growing role of data science and computational modeling methods in identifying drugs with potential for repositioning. Certain business aspects of drug innovation, especially the relevant factors of market exclusivity, are also discussed. Despite the promising potential of drug repositioning for innovation, it remains underutilized, especially in veterinary applications. To change this landscape for mutual benefits of human and veterinary drug innovation, further exploitation of the potency of drug repositioning is necessary through closer cooperation between all stakeholders, academia, industry, pharmaceutical authorities, and innovation policy makers, and the integration of human and veterinary repositioning into a unified innovation space. For this purpose, the establishment of the conceptually new “One Health Drug Repositioning Platform” is proposed. Oncology is one of the disease areas where this platform can significantly support the development of new drugs for human and dog (or other companion animals) anticancer therapies. As an example of the utilization of human and veterinary drugs for veterinary repositioning, the use of COX inhibitors to treat dog cancers is reviewed.