Various types of additive manufacturing (AM) methods (also called 3D printing), and materials have been increasingly studied in the field of additive manufacturing of flexible structures such as fabrics, and flexible electronics. Polymer-based AM processes allow the flexibility, rapid, and low-cost fabrication of complex geometries depending on the types of materials used. The purpose of this review article is to summarize the major AM methods, materials, and their emerging applications to additively manufacture the flexible structures. In the AM methods section, Fused Deposition Modeling (FDM), and Selective Laser Sintering (SLS) are reviewed for fabrics, and Direct Ink Writing (DIW) for electronics. In the Materials section, the manufacturing methods, chemical structures, properties, advantages, and limitations of some of the widely used materials in three-dimensional (3D) printing of polymers are reviewed. Third, the applications of these methods and materials for fabrics, and electronics are covered. Finally, the associated opportunities and challenges in 3D printing process of flexible structures are described. The future research should be related to the exploration of combinations and development of innovative materials, printing process parameters, detail study on improving the properties, and hybrid 3D printing process.