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. Display omitted •RNA & DNA SPRITE comprehensively maps the spatial organization of RNA and DNA•Hundreds of ncRNAs form high-concentration territories throughout the nucleus•ncRNAs recruit diffusible RNA and protein regulators into precise 3D structures•ncRNA compartments can shape DNA contacts, heterochromatin, and gene expression Mapping the proximity of RNAs to DNA and to other RNAs elucidates how nuclear non-coding RNAs serve as spatial organizers controlling processes underpinning gene regulation.