Calcium phosphate (CaP) materials are important inorganic constituents in biological hard tissue. CaPs, including amorphous calcium phosphate (ACP) have been widely applied in dental and biomedical applications, such as tissue engineering. Scaffold constructs are commonly used as templates to create a biomimetic environment. This review considers ACP scaffold fabrication techniques, including tissue-engineered constructs with intrinsic incorporation of ACP as well as scaffolds formed via precipitation of mineralized solutions on a substrate. Attention is given to the approaches used to assess cellular and molecular responses elicited by ACP scaffolds, such as biocompatibility, cell conductivity, cell adhesion, cell differentiation, phenotypic profiles, and gene expression. Bioactivity of composite ACP scaffolds can be enhanced by incorporating biomolecules to create multi-functional properties. Herein we summarize the use of antibiotics, growth factors, and gene delivery systems to create multi-functional ACP scaffolds. Inasmuch as CaP materials have been investigated as drug delivery systems for many years, we briefly consider the potential of integrating these systems with existing ACP scaffold constructs and the potential for precision medicine.