Topological Quantum Chemistry is a new framework that unifies real and momentum space descriptions of solids, providing a complete and predictive theory for classifying topological insulators (TIs) and semimetals. The approach combines group theory and graph theory to analyze the global structure of electronic band structures, revealing the connection between topology and local chemical bonding. By considering all 230 crystal symmetry groups, the theory identifies which band structures are topologically nontrivial and how they relate to the chemical orbitals at the Fermi level. It classifies all possible band structures for each symmetry group and determines which are topologically trivial or nontrivial. The theory also introduces the concept of elementary band representations (EBRs), which are the smallest sets of bands derived from local atomic-like Wannier functions. The number of EBRs at the Fermi level determines the topological classification of materials. The method allows for the prediction of new TIs and semimetals by analyzing the connectivity of bands in momentum space and their transformation properties in real space. The approach is validated by identifying known TIs and predicting hundreds of new ones. The theory also provides a systematic way to search for TIs by analyzing the symmetry and orbital structure of materials. The results show that topological materials are not inherently rare but rather result from the constraints imposed by crystal symmetries and the topology of band structures. The work demonstrates the power of combining group theory, chemistry, and graph theory to understand and predict the topological properties of materials.Topological Quantum Chemistry is a new framework that unifies real and momentum space descriptions of solids, providing a complete and predictive theory for classifying topological insulators (TIs) and semimetals. The approach combines group theory and graph theory to analyze the global structure of electronic band structures, revealing the connection between topology and local chemical bonding. By considering all 230 crystal symmetry groups, the theory identifies which band structures are topologically nontrivial and how they relate to the chemical orbitals at the Fermi level. It classifies all possible band structures for each symmetry group and determines which are topologically trivial or nontrivial. The theory also introduces the concept of elementary band representations (EBRs), which are the smallest sets of bands derived from local atomic-like Wannier functions. The number of EBRs at the Fermi level determines the topological classification of materials. The method allows for the prediction of new TIs and semimetals by analyzing the connectivity of bands in momentum space and their transformation properties in real space. The approach is validated by identifying known TIs and predicting hundreds of new ones. The theory also provides a systematic way to search for TIs by analyzing the symmetry and orbital structure of materials. The results show that topological materials are not inherently rare but rather result from the constraints imposed by crystal symmetries and the topology of band structures. The work demonstrates the power of combining group theory, chemistry, and graph theory to understand and predict the topological properties of materials.