The textile and clothing (T&C) industry is not usually viewed as an exemplar of sustainable development and the circular economy (CE), as the industry has hitherto developed its products in a linear fashion, with relatively little recycling of the finished goods. This article examines the industry’s approach to the core sustainability concept and the CE in particular through a review of the available academic literature, evidence from corporate sustainability reports and websites, and feedback from an online survey of industry professionals. More specifically, the article investigates how German T&C companies are addressing sustainability in their corporate strategies and what activities relating to the CE are being pursued in the industry. The role of digital technologies in the transition to sustainability and the CE in the German T&C industry is also explored. The study finds that whilst sustainability is now firmly embedded at the strategic level in the vast majority of the companies studied, attitudes towards the CE are mixed. The use of digital technologies in support of sustainability objectives is also limited at present, but the need to meet compliance requirements and new customer perceptions of sustainability will speed the transition to CE activities, which will be facilitated by the greater exploitation of these technologies. An operational framework for initiating such a transition is developed, and action lists in the key areas of change—organization, products, and processes—are presented. These may be used as a guideline for practitioners, and the findings also make a small contribution to the scarcity of literature in this field of research.