The DNA of eukaryotic chromatin and chromosomes is repeatedly supercoiled around histone octamers forming ‘beads-on-a-string’ chains of nucleosomes. The extent of nucleosome chain folding and DNA accessibility vary between different functional and epigenetic states of nuclear chromatin and change dramatically upon cell differentiation, but the molecular mechanisms that direct 3D folding of the nucleosome chain in vivo are still enigmatic. Recent advances in cell imaging and chromosome capture techniques have radically challenged the established paradigm of regular and hierarchical chromatin fibers by highlighting irregular chromatin organization and the importance of the nuclear skeletal structures hoisting the nucleosome chains. Here, we argue that, by analyzing individual structural elements of the nucleosome chain – nucleosome spacing, linker DNA conformations, internucleosomal interactions, and nucleosome chain flexibility – and integrating these elements in multiplex 3D structural models, we can predict the features of the multiplex chromatin folding assemblies underlying distinct developmental and epigenetic states in living cells. Furthermore, partial disassembly of the nuclear structures suspending chromatin fibers may reveal the intrinsic mechanisms of nucleosome chain folding. These mechanisms and structures are expected to provide molecular cues to modify chromatin structure and functions related to developmental and disease processes.