In eukaryotes, the genome is packed into chromosomes, each consisting of large polymeric fibers made of DNA bound with proteins (mainly histones) and RNA molecules. The nature and precise 3D organization of this fiber has been a matter of intense speculations and debates. In the emerging picture, the local chromatin state plays a critical role in all fundamental DNA transactions, such as transcriptional control, DNA replication or repair. However, the molecular and structural mechanisms involved remain elusive. The purpose of this review is to give an overview of the tremendous efforts that have been made for almost 40 years to build physiologically relevant models of chromatin structure. The motivation behind building such models was to shift our representation and understanding of DNA transactions from a too simplistic 'naked DNA' view to a more realistic 'coated DNA' view, as a step towards a better framework in which to interpret mechanistically the control of genetic expression and other DNA metabolic processes. The field has evolved from a speculative point of view towards in vitro biochemistry and in silico modeling, but is still longing for experimental in vivo validations of the proposed structures or even proof of concept experiments demonstrating a clear role of a given structure in a metabolic transaction. The mere existence of a chromatin fiber as a relevant biological entity in vivo has been put into serious questioning. Current research is suggesting a possible reconciliation between theoretical studies and experiments, pointing towards a view where the polymorphic and dynamic nature of the chromatin fiber is essential to support its function in genome metabolism.