Identification of relevant structures and reaction pathways in ever more complex composite materials and nanostructures poses a central challenge to computational materials discovery. Efficient, unbiased global structure prediction, tailored to identify chemically relevant domains of compound space, is an important tool to achieve rational materials design. In this talk, I discuss a class of curvilinear, internal coordinate systems with a long history of application in chemistry in a different perspective, namely global materials structure search. Curvilinear topologies have a structural descriptive power with interesting properties that seamlessly connect to modern concepts of machine learning and graph theory. At hand of examples, I will show that they provide the means to efficient structure determination of clusters, complex covalent molecules, and hybrid organic-inorganic interfaces, while at the same time requiring only minor modification of existing algorithms. Using machine-learning concepts, their success in structure search, pattern recognition, and potential energy interpolation can be
rationalized in the correct representation of molecular connectivity and structural bonding patterns.