Additive manufacturing is a highly effective and versatile technology, especially in the medical sector, due to its customization, material complexity, design flexibility, waste minimization, and ability to fabricate intricate shapes that are cumbersome to manufacture by conventional manufacturing techniques. 4D printing plays a significant role in the medical field, especially in the areas not covered by 3D printing technologies, such as smart implants, devices and tools. Also, 4D printing helps doctors to treat more patients with high accuracy and quality. Hence, this manuscript aims to provide an overview of distinct 3D and 4D printing techniques and their emerging applications in the medical sector. A study of 3D printing technologies is presented by explaining the working principles of distinct 3D printing methods: stereo lithography, fusion deposition modeling, inkjet printing, selective laser sintering, selective laser melting and electron beam melting. In addition, the emerging applications of 3D printing in medical sectors (e.g., bioprinting, surgical guides, pharmaceuticals, prostheses, medical devices, dentistry, physiotherapy, etc.), as well as challenges and the future scope of 3D printing, are also discussed. Further, the concept of 4D printing, the market for both 3D and 4D printing, the benefits of 4D printing, the comparison of 3D and 4D printing, limitations, applications, and the future scope of 4D printing in the medical sector are also covered.