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The Universe evolves as it expands. It was once generally smooth and filled with neutral gas, as revealed by the cosmic microwave background observations. Today’s Universe is predominantly filled with ionised plasmas threaded by magnetic fields. How did the transition happen? How do we probe the cosmic evolutionary history from the Earth? In this colloquium, I will discuss how astrophysicists explore the evolving Universe, with foci on two research frontiers: large-scale cosmic magnetic fields and gas reionisation. These two science themes have formed several key science projects for the forthcoming unprecedentedly powerful radio telescopes, the Square Kilometre Array (SKA), its precursors (e.g. ASKAP, MeerKAT, MWA) and pathfinders (e.g. LOFAR), among many other experiments. I will highlight, with demonstrations, the essential ingredients to optimise the scientific gains from these observational experiments by studying how information is encoded into the radiation we receive (i.e., cosmological radiative transfer). Finally, I will conclude this talk with a summary of the key findings from my theoretical calculations and an outlook on the exciting developments in our understanding of the ever-changing cosmos.