With the development of 3D, 4D, and 5D printing technology, the field of additive manufacturing has undergone tremendous change. This chapter explores the principles and current developments in these fields with an emphasis on polymers and their nanocomposites. It examines how advances in multi-material and multi-scale 4D printing have transformed a range of industries, including aerospace and healthcare, and highlights how these technologies can be used to build intelligent, shape-shifting objects that are capable of self-assembly. Additionally, the notion of 5D printing is introduced, highlighting its potential for dynamic applications in biomedicine, robotics, and sensors. 5D printing includes time as an additional dimension. Issues with material compatibility, resolution, and post-processing are discussed, in addition to sustainable methods and choices for biodegradable polymers. Incorporating nanoparticles to improve material qualities including strength, conductivity, and thermal stability is also covered in this chapter. Lastly, it offers a futuristic perspective on the widespread industrial use of 3D, 4D, and 5D printing of polymers and their nanocomposites, promoting innovation and sustainability in industries like electronics, personalized medicine, and architecture. For academics, engineers, and innovators working in the ever-changing field of polymer additive manufacturing, this chapter provides a thorough reference.