Magnetic fields are ubiquitous on astrophysical and cosmological scales: from planets and stars to galaxies and galaxy clusters. Different observational methods infer a field strength of the order of microGauss and coherence scales reaching a few tens of kiloparsecs in galaxy clusters. Despite their ubiquity, the origin of these fields still remains unknown. It is commonly assumed that the observed fields are originated from either astrophysical or cosmological (primordial) weak seed magnetic fields that undergo efficient growth during structure formation. The widespread presence of magnetic fields in high redshift galaxy clusters as well as observations of blazar spectra tip the scale in favour of the primordial magnetogenesis scenario. Primordial Magnetic Fields (PMFs), i.e., seed magnetic fields generated in the early Universe, could come from inflation and phase-transitions epochs. In inflationary magnetogenesis, the coherence scales of PMFs can be stretched on superhorizon scales while the coherence scale of phase transition-generated fields is bounded by the Hubble horizon at the moment of the field generation.  In our work, we study the evolution of inflationary and phase transition-generated PMFs in the cosmic web and galaxy clusters. We employ the cosmological MHD code ENZO to study their evolution during structure formation while taking into account their pre-recombination decay dynamics (that has been studied previously with the PENCIL code). In this talk, I will discuss the results of these cosmic web and galaxy cluster studies that argue in favour of distinguishing between different PMFs on galaxy-cluster as well as on filamentary and cosmic voids’ scales. I will also discuss the efficiency of turbulent amplification of different PMFs on galaxy cluster scales and present our constraints on the field strength based on the Faraday rotation analysis. Finally, I will present the implications of our work in the context of current/future blazar spectra observations.