Chromosomes in proliferating metazoan cells undergo marked structural metamorphoses every cell cycle, alternating between highly condensed mitotic structures that facilitate chromosome segregation, and decondensed interphase structures that accommodate transcription, gene silencing and DNA replication. Here we use single-cell Hi-C (high-resolution chromosome conformation capture) analysis to study chromosome conformations in thousands of individual cells, and discover a continuum of cis-interaction profiles that finely position individual cells along the cell cycle. We show that chromosomal compartments, topological-associated domains (TADs), contact insulation and long-range loops, all defined by bulk Hi-C maps, are governed by distinct cell-cycle dynamics. In particular, DNA replication correlates with a build-up of compartments and a reduction in TAD insulation, while loops are generally stable from G1 to S and G2 phase. Whole-genome three-dimensional structural models reveal a radial architecture of chromosomal compartments with distinct epigenomic signatures. Our single-cell data therefore allow re-interpretation of chromosome conformation maps through the prism of the cell cycle.