Chromatin packing in eukaryotic chromosomes has been traditionally viewed as a hierarchical process, in which nucleosome chains fold into helical chromatin fibers. These fibers would then fold into more complex regular structures. However, recent chromatin imaging studies and analyses of chromosomal DNA contacts within the 3D space of the cell nucleus have necessitated a radical revision of the hierarchical chromatin packing model. According to the new studies, the nucleosome chain has a free spatial configuration without regular helical fibers in most cell types. The overall 3D organization of DNA in the cell nucleus includes chromatin loops and contact domains of up to several million base pairs in size. During cell differentiation, individual structure-functional chromatin domains marked by similar types of histone modifications and functional states can merge together and form chromosomal subcompartments suited for local gene activation or repression. This "attraction of likenesses" may be mediated by direct self-association of nucleosome chains as well as by architectural chromatin proteins making oligomeric protein "bridges" between nucleosomes as well as larger dynamic condensates leading to liquid-liquid phase separation inside the cell nucleus. Future studies of mechanisms of chromatin self-association and compartmentalization will require a combination of molecular, imaging, and computational approaches capable of revealing the 3D organization of the eukaryotic genome with nucleosomal resolution.