Chemical compound space (CCS) is the combinatorial set which encompasses all chemical compounds. It can be viewed as the high dimensional space spanned by all the possible stoichiometries and configurations of electrons and atomic nuclei which form molecular or condensed matter. Due to the combinatorial nature of CCS, systematic screening for interesting properties or even simple enumeration is beyond any computational capacity. But CCS provides a natural framework in which to construct rigorous mathematical tools for the development of direct and inverse quantitative structure-property relationships, which can be applied to challenges in Materials and Bio design. Diverse scientific areas are involved, which benefit from historically grown experimental insights as well as advances made in theoretical and computational sciences. They include statistical mechanics, liquid and solid state physics, quantum chemistry, graph theory, molecular physics, condensed matter physics, optimization algorithms, data mining, statistical analysis, and others.