The discovery of gravitational waves has opened a new era of astronomy. If generated by physical processes in the early Universe, these waves reach us today as a stochastic background. Ongoing and planned missions aim to probe this background across a wide range of frequency bands, providing a unique window into physical processes occurring at different cosmic epochs. These processes typically involve length scales far smaller than those accessible through traditional electromagnetic observations.

In this talk, I will present some of our recent work on three key eras of the early Universe. First, I will discuss the epoch of inflation, focusing on a well-motivated model and going beyond the homogeneous and linear treatment by presenting results from our lattice simulations, along with their implications. Second, I will address the generation of observed large-scale magnetic fields in the early Universe and how the associated stochastic gravitational wave background can be used to probe such scenarios. Finally, I will briefly describe our ongoing efforts to simulate first-order phase transitions, study their hydrodynamics, and obtain more accurate gravitational wave spectra, which may help distinguish them from other physical processes.