In this thesis I present some results from my research into eukaryotic chromatin structure and dynamics. Chromosome organisation is studied through a simulation approach and at different stages of the cell cycle. Throughout all the thesis, simulations are carried out by using "beads-on-a-string" polymer models whose details are given in Chapter 2. Depending on the aim of simulations, different details of chromatin fibres and of the cell environment are included in the model, in order to reproduce basic characteristics of chromosomes in that particular phase of the cell cycle. Chapter 3 focuses on the study of chromatin dynamics during interphase when different chromosomes are not experimentally distinguishable, and chromatin is organised in topologically associating domains (TADs) and active and inactive compartments (respectively referred to as A and B compartments in the literature). Within this scenario, I depict the Pax6 gene locus by employing the HiP-HoP polymer model which has already been applied to reproduce interactions of Pax6 in mouse cell lines and of SOX2 in human cell lines. DNA accessibility data (ATAC-seq), ChIP-seq data for CTCFs and cohesins, and ChIP-seq data for H3K27ac marks are included to model the fibre's structure and to account for proper interactions with transcription factors. After validating the model by comparing simulation results with experimental FISH and CaptureC data, I show the significant correlation between gene expression and chromosome mobility inside the nucleus. Within interphase, a particular important step is the S phase during which DNA synthesis happens. At this stage DNA polymerases bind DNA filaments forming large structures called replication factories which synthesise a new DNA molecule. Fluorescence microscopy experiments have shown an interesting dynamics of replication factories which, during S phase, increase their size and decrease in number. To understand such a singular dynamics, in Chapter 4 I develop a polymer physics model to study the dynamics of DNA (more precisely chromatin) replication. I will show that this model naturally leads to the self-organisation of chromatin into clusters, or replication factories, whose dynamics may entail different regimes according to the microscopic rules underlying the model. Important factors determining the emerging behaviour of the system are the interactions between proteins and chromatin, and the interplay between the directed motion of DNA polymerases along chromatin in 1D and their diffusion in 3D. Finally, I study chromosome organisation during mitosis. At this stage TADs and A and B compartments disappear, while chromosomes condensate into cylindrical structures becoming distinguishable through microscopy techniques. Experiments suggest that late prophase chromosomes are organised as arrays of consecutive loops which might be originated by a loop extrusion process performed by condensins II acting as molecular motors. In Chapter 5 I show a simple attractive interaction between bottle brush chromosomes and protein complexes (for example condensins I) can lead to the shorter and thicker mitotic cylinders observed during prometaphase without requiring the mediation of additional molecular motors.