Thermoplastic elastomer (TPE), one of the most important classes of specialty polymeric materials, has immense potential in several industrial applications. They have unique rubber like elasticity, thermoplastics like processability and recyclability. Compared to the conventional processing methods, material extrusion 3D printing based on fused deposition modeling (FDM) is an advanced and highly sophisticated processing technology which allows rapid prototyping and mass customization and is an ideal processing method for thermoplastic elastomeric materials to fabricate several complex computerized geometrics. However, FDM 3D printing of thermoplastic elastomeric materials is currently in the development stage and is facing several challenges including limited FDM compatible printable TPEs, difficulties in printability of soft TPEs, several defects in printed parts and poor mechanical properties. The objective of this review article is to figure out the ideas to address these challenges. It focuses on how to develop FDM compatible new thermoplastic elastomeric materials and how to make unprintable commercially available TPEs printable. The state-of-the-art advancements in FDM 3D printing of thermoplastic elastomeric materials are summarized. Strategies to overcome the defects in 3D printed TPE parts and to improve mechanical and thermoplastic elastomeric properties of 3D printed TPEs are provided. Finally, further prospective for this rapidly progressing field are critically discussed to motivate potential research in this field. It is expected that this review will be useful for both the academia and industries for any research work involved with the FDM of thermoplastic elastomeric materials.