Ever since its introduction over five decades ago, geometric solid modeling has been crucial for engineering design purposes and is used in engineering software packages such as computer-aided design (cad), computer-aided manufacturing, computer-aided engineering, etc. Solid models produced by cad software have been used to transfer geometric information from designers to manufacturers. Since the emergence of additive manufacturing (AM), a CAD file can also be directly uploaded to a three-dimensional (3D) printer and used for production. AM techniques allow manufacturing of complex geometric objects such as bio-inspired structures and lattice structures. These structures are shapes inspired by nature and periodical geometric shapes consisting of struts interconnecting in nodes. Both structures have unique properties such as significantly reduced weight. However, geometric modeling of such structures has significant challenges due to the inability of current techniques to handle their geometric complexity. This calls for a novel modeling method that would allow engineers to design complex geometric objects. This survey paper reviews geometric modeling methods of complex structures to support bio-inspired design created for AM which includes discussing reasoning behind bio-inspired design, limitations of current modeling approaches applied to bio-inspired structures, challenges encountered with geometric modeling, and opportunities that these challenges reveal. Based on the review, a need for a novel geometric modeling method for bio-inspired geometries produced by AM is identified. A framework for such a bio-inspired geometric modeling method is proposed as a part of this work.