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Structural characterization of polymer materials is a major step in the process of creating materials' design-structural-property relationships. With growing interests in artificial intelligence (AI)-driven materials design and high-throughput synthesis and measurements, there is now a critical need for development of complementary data-driven approaches (e.g., machine learning models and workflows) to enable fast and automated interpretation of the characterization results. This review sets out with a description of the needs for machine learning specifically in the context of three commonly used structural characterization techniques for polymer materials: microscopy, scattering, and spectroscopy. Subsequently, a review of notable work done on development and application of machine learning models / workflows for these three types of measurements is provided. Definitions are provided for common machine learning terms to help readers who may be less familiar with the terminologies used in the context of machine learning. Finally, a perspective on the current challenges and potential opportunities to successfully integrate such data-driven methods in parallel/sequentially with the measurements is provided. The need for innovative interdisciplinary training programs for researchers regardless of their career path/employment in academia, national laboratories, or research and development in industry is highlighted as a strategy to overcome the challenge associated with the sharing and curation of data and unifying metadata.