Additive manufacturing (AM), commonly referred to as 3D printing, has revolutionized the manufacturing landscape by enabling the intricate layer-by-layer construction of three-dimensional objects. In contrast to traditional methods relying on molds and tools, AM provides the flexibility to fabricate diverse components directly from digital models without the need for physical alterations to machinery. Four-dimensional printing is a revolutionary extension of 3D printing that introduces the dimension of time, enabling dynamic transformations in printed structures over predetermined periods. This comprehensive review focuses on polymeric materials in 3D printing, exploring their versatile processing capabilities, environmental adaptability, and applications across thermoplastics, thermosetting materials, elastomers, polymer composites, shape memory polymers (SMPs), including liquid crystal elastomer (LCE), and self-healing polymers for 4D printing. This review also examines recent advancements in microvascular and encapsulation self-healing mechanisms, explores the potential of supramolecular polymers, and highlights the latest progress in hybrid printing using polymer–metal and polymer–ceramic composites. Finally, this paper offers insights into potential challenges faced in the additive manufacturing of polymer composites and suggests avenues for future research in this dynamic and rapidly evolving field.