The study of galaxy formation and evolution is a humbling exercise of acknowledging the monumental discrepancy between human lifespans and cosmic timescales and distances. We can observe how galaxies have formed and evolved from when the Universe was in its infancy to the present day, establishing connections between galaxies across cosmic time. Galaxies are complex systems made out of gas, dust, stars, and dark matter, and all but the latter emit radiation that we can detect and interpret. By observing the light from galaxies, we can infer their physical properties, characterising each component, and understanding the processes that take place in the largest chemistry and physics laboratories in the Universe.

This thesis embarks on a comprehensive exploration of galaxies, from 1 Gyr after the Big Bang to today, using the Hubble and James Webb Space Telescopes. A significant gap emerges in the study of galaxies when contrasting the local and the early Universe. Historically, high-redshift galaxies have been hindered by limited sensitivity and spatial resolution, which restricts our ability to resolve their components and conduct detailed studies of their internal structure and diversity of their stellar populations. On the other hand, local studies are enriched with an abundance of data and information, allowing us to characterise them even down to scales of tens of parsecs of individual star-forming clouds. The central theme of this thesis resolves this disparity in the study of galaxies throughout 13 billion years of cosmic time. I use spatially-resolved modelling techniques to infer the physical properties of galaxies on a pixel-by-pixel basis.

The first part of the thesis focuses in a sample of 24 local star-forming galaxies observed with the Hubble Space Telescope at z ∼ 0. By developing methods to extract robust emission line fluxes from narrow-band photometry, I investigate obscured star forming regions with rarely used hydrogen recombination lines. The Paschen-series line emission probes systematically higher star formation rates than the commonly used Balmer Hα line, suffering less obscuration. With observations of Paschen lines at high redshift now enabled by JWST, we can better determine the star formation history in the z > 1 Universe produced by dusty star-forming galaxies.

In the second part I present the implications that arise from studying highredshift galaxies in a resolved approach now possible with JWST, in contrast with the broadly adopted simplified and unresolved methods. Surprisingly, the inferred stellar masses are heavily impacted by this, potentially challenging current galaxy and cosmology models. In a sample of five galaxies at 5 < z < 9, I find that stellar masses can be underestimated by up to a factor of ten when not resolved, given the outshining by the youngest stellar populations that can dominate the integrated light of the galaxy. Recognizing the limitations of photometriconly studies, the last work combines spectroscopic information to demonstrate that our methodology is robust. I explore the contribution of a strongly-lensed galaxy to the end of reionisation, finding large variations of the ionising photon production efficiency across the galaxy.

This thesis provides valuable insights into the intricate evolution of galaxies and highlights the importance of spatially-resolved analyses to understand galaxies as the complex systems that they are.