RNA, DNA, and protein molecules are highly organized within three-dimensional (3D) structures in the nucleus. Although RNA has been proposed to play a role in nuclear organization, exploring this has been challenging because existing methods cannot measure higher-order RNA and DNA contacts within 3D structures. To address this, we developed RNA & DNA SPRITE (RD-SPRITE) to comprehensively map the spatial organization of RNA and DNA. These maps reveal higher-order RNA-chromatin structures associated with three major classes of nuclear function: RNA processing, heterochromatin assembly, and gene regulation. These data demonstrate that hundreds of ncRNAs form high-concentration territories throughout the nucleus, that specific RNAs are required to recruit various regulators into these territories, and that these RNAs can shape long-range DNA contacts, heterochromatin assembly, and gene expression. These results demonstrate a mechanism where RNAs form high-concentration territories, bind to diffusible regulators, and guide them into compartments to regulate essential nuclear functions.