Through the development of additive manufacturing (AM), significant progress has been made in the field of biomedical devices and bone tissue engineering. AM allows for the fabrication of site-specific implants with tailorable porosity and controlled chemistry through the utilization of innovative materials. With the development of composite scaffolds, valiant efforts have been made in relation to the biosignificance of synthetic bone grafting materials. This paradigm shift is only made possible by the discovery of new and novel compounds for scaffold preparation. There is a growing clinical interest regarding natural polymers in order to generate novel properties and functionalities in biomaterial applications. Through 3D printing (3DP) and AM techniques, natural polymer incorporation into bioceramic-based scaffolds and hydrogels has allowed for the advanced treatment of bone ailments and defects through the creation of novel, functionalized composite surfaces. Naturally sourced polymers including chitosan, alginate, collagen, gelatin, cellulose, hyaluronate, silk, fibrinogen, and starch show great potential in the biomedical field due to their inherent biocompatibility, bioactivity, and bioresorbability. This review paper will introduce these natural biopolymers alongside their traditional biomedical applications followed by traditional processing techniques of these biopolymers in conjunction with calcium phosphate in the manufacturing of scaffolds that can be used in the bone tissue engineering field. The final section of the review focuses on the advancement of AM of natural polymers with an emphasis on their applications as scaffolds and hydrogels also for utilization in the bone tissue engineering field.