The DNA replication process needs to be coordinated with other DNA metabolism transactions and must eventually extend to the full genome, regardless of chromatin status, gene expression, secondary structures and DNA lesions. Completeness and accuracy of DNA replication are crucial to maintain genome integrity, limiting transformation in normal cells and offering targeting opportunities for proliferating cancer cells. DNA replication is thus tightly coordinated with chromatin dynamics and 3D genome architecture, and we are only beginning to understand the underlying molecular mechanisms. While much has recently been discovered on how DNA replication initiation is organised and modulated in different genomic regions and nuclear territories-the so-called "DNA replication program"-we know much less on how the elongation of ongoing replication forks and particularly the response to replication obstacles is affected by the local nuclear organisation. Also, it is still elusive how specific components of nuclear architecture participate in the replication stress response. Here, we review known mechanisms and factors orchestrating replication initiation, and replication fork progression upon stress, focusing on recent evidence linking genome organisation and nuclear architecture with the cellular responses to replication interference, and highlighting open questions and future challenges to explore this exciting new avenue of research.