Block copolymers are extremely versatile materials that microphase separate to give rise to a rich array of complex behavior, making them the ideal platform for the development of rheologically sophisticated soft matter. In line with growing environmental concerns of conventional plastics from petroleum feedstocks, this work focuses on the rheological design of sustainable block copolymers - those derived from renewable sources and are degradable - based on poly(lactide). Although commercially viable, poly(lactide) has a number of inherent deficiencies that result in a host of challenges that require both creative and practical solutions that are cost-effective and amenable to large-scale production. Specifically, this dissertation looks at applications in which both shear and extensional rheology dictate performance attributes, namely chewing gum, pressure-sensitive adhesives, and polymers for blown film extrusion. Structure-property relationships in the context of block polymer architecture, polymer composition, morphology, and branching are explored in depth. The basic principles and fundamental findings presented in this thesis are applicable to a broader range of substances that incorporate block copolymers for which rheology plays a pivotal role.