The mechanism by which the "beads-on-a-string" nucleosome chain folds into various higher-order chromatin structures in eukaryotic cell nuclei is still poorly understood. The various models depicting higher-order chromatin as regular helical fibers and the very opposite "polymer melt" theory imply that interactions between nucleosome "beads" make the main contribution to the chromatin compaction. Other models in which the geometry of linker DNA "strings" entering and exiting the nucleosome define the three-dimensional structure predict that small changes in the linker DNA configuration may strongly affect nucleosome chain folding and chromatin higher-order structure. Among those studies, the cross-disciplinary approach pioneered by Jörg Langowski that combines computational modeling with biophysical and biochemical experiments was most instrumental for understanding chromatin higher-order structure in vitro. Strikingly, many recent studies, including genome-wide nucleosome interaction mapping and chromatin imaging, show an excellent agreement with the results of three-dimensional computational modeling based on the primary role of linker DNA geometry in chromatin compaction. This perspective relates nucleosome array models with experimental studies of nucleosome array folding in vitro and in situ. I argue that linker DNA configuration plays a key role in determining nucleosome chain flexibility, topology, and propensity for self-association, thus providing new implications for regulation of chromatin accessibility to DNA binding factors and RNA transcription machinery as well as long-range communications between distant genomic sites.