Over millions of years of evolution, nature has created organisms with overwhelming performances due to their unique materials and structures, providing us with valuable inspirations for the development of next-generation biomedical devices. As a promising new technology, 3D printing enables the fabrication of multiscale, multi-material, and multi-functional three-dimensional (3D) biomimetic materials and structures with high precision and great flexibility. The manufacturing challenges of biomedical devices with advanced biomimetic materials and structures for various applications were overcome with the flourishing development of 3D printing technologies. In this paper, the state-of-the-art additive manufacturing of biomimetic materials and structures in the field of biomedical engineering were overviewed. Various kinds of biomedical applications, including implants, lab-on-chip, medicine, microvascular network, and artificial organs and tissues, were respectively discussed. The technical challenges and limitations of biomimetic additive manufacturing in biomedical applications were further investigated, and the potential solutions and intriguing future technological developments of biomimetic 3D printing of biomedical devices were highlighted.