As the digital era advances, many industries continue to expand their use of digital technologies to support company operations, notably at the customer interface, bringing new commercial opportunities and increased efficiencies. However, there are new sets of responsibilities associated with the deployment of these technologies, encompassed within the emerging concept of corporate digital responsibility (CDR), which to date has received little attention in the academic literature. This exploratory paper thus looks to make a small contribution to addressing this gap in the literature. The paper adopts a qualitative, inductive research method, employing an initial scoping literature review followed by two case studies. Based on the research findings, a simple model of CDR parameters is put forward, and the article concludes with a discussion of a number of emergent issues - fair and equitable access, personal and social well-being, environmental implications, and cross-supply chain complexities - that suggest possible directions for future research.