Additive manufacturing (AM), sometimes called three-dimensional (3D) printing, has attracted a lot of research interest and is presenting unprecedented opportunities in biomedical fields, because this technology enables the fabrication of biomedical constructs with great freedom and in high precision. An important strategy in AM of biomedical constructs is to mimic the structural organizations of natural biological organisms. This can be done by directly depositing cells and biomaterials, depositing biomaterial structures before seeding cells, or fabricating molds before casting biomaterials and cells. This review organizes the research advances of AM-based biomimetic biomedical constructs into three major directions: 3D constructs that mimic tubular and branched networks of vasculatures; 3D constructs that contains gradient interfaces between different tissues; and 3D constructs that have different cells positioned to create multicellular systems. Other recent advances are also highlighted, regarding the applications of AM for organs-on-chips, AM-based micro/nanostructures, and functional nanomaterials. Under this theme, multiple aspects of AM including imaging/characterization, material selection, design, and printing techniques are discussed. The outlook at the end of this review points out several possible research directions for the future.