Sustainability is not a one-player task. Organizations have started to realize that their supply chains have a significant social and environmental impact, usually greater than their own operations, and managing sustainability at suppliers is crucial for supply chain-wide sustainability. Supplier sustainability management (SSM) research is fast evolving across multiple disciplines but lacks an interdisciplinary review to guage the progress made, and to decide the path forward. Heightened global focus on sustainability compels us to explore research avenues in SSM for meaningful progress. In this paper, we provide a comprehensive review of SSM research including the most recent work. We propose the Motivation-Measure-Analyze-Improve-Govern (MMAIG) framework for supplier sustainability, identify the limitations of current SSM research in enabling this framework, and propose future research directions. Our key observations are that (a) current SSM research is heavily focused on measuring and monitoring supplier sustainability, and (b) supplier sustainability improvement/development research is limited and the majority of it is about sustainable supplier selection. The future research directions that we propose are centered around (a) optimizing the investments towards supplier sustainability through collaboration, proposing mechanisms that consider risks, liabilities, and gains of all parties, and (b) considering behavioral aspects to overcome SSM implementation issues. Organizations can achieve efficient improvement in supplier sustainability by using a collaborative approach that is data-driven and trust-based. We discuss several mechanisms within our MMAIG framework that can help organizations in their collaborative approach.