4OR-Q J Oper Res (2016) 14:337–376 DOI 10.1007/s10288-016-0314-2 INVITED SURVEY Assigned and unassigned distance geometry: applications to biological molecules and nanostructures Simon J. L. Billinge 1,2 · Phillip M. Duxbury 3 · Douglas S. Gonçalves 4 · Carlile Lavor 5 · Antonio Mucherino 6 Received: 19 February 2016 / Published online: 4 April 2016 © Springer-Verlag Berlin Heidelberg 2016 Abstract Considering geometry based on the concept of distance, the results found by Menger and Blumenthal originated a body of knowledge called distance geometry. This survey covers some recent developments for assigned and unassigned distance geometry and focuses on two main applications: determination of three-dimensional conformations of biological molecules and nanostructures. B Douglas S. Gonçalves douglas.goncalves@ufsc.br Simon J. L. Billinge sb2896@columbia.edu Phillip M. Duxbury Duxbury@pa.msu.edu Carlile Lavor clavor@ime.unicamp.br Antonio Mucherino antonio.mucherino@irisa.fr 1 Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA 2 X-ray Scattering Group, Brookhaven National Laboratory, Upton, NY 11973, USA 3 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA 4 Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis, Brazil 5 Department of Applied Mathematics (IMECC-UNICAMP), University of Campinas, Campinas, SP 13081-970, Brazil 6 Institut de Recherche en Informatique et Systèmes Aléatoires, Université de Rennes 1, 35042 Rennes, France 123