Chromosomes in proliferating metazoan cells undergo dramatic structural metamorphoses every cell cycle, alternating between highly condensed mitotic structures facilitating chromosome segregation, and decondensed interphase structures accommodating transcription, gene silencing and DNA replication. Here we use single-cell Hi-C 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 build-up of compartments and reduction in TAD insulation, while loops are generally stable from G1 through S and G2. Whole-genome 3D structural models reveal a radial architecture of chromosomal compartments with distinct epigenomic signatures. Our single-cell data thereby allow for re-interpretation of chromosome conformation maps through the prism of the cell cycle.