Abstract: The direct observation of gravitational waves provides a powerful new lens with which to explore the cosmos. In this talk, I will share recent observational developments illustrating how these measurements can be used to not only give tantalising clues into our early Universe, but also pin down new physics far beyond reach of laboratory experiments. One particularly fruitful avenue, is the search for gravitational waves from cosmological phase transitions. In this talk I will explore what is required for these transitions to be observable (leading to a class of theories exhibiting a phenomenon called supercooling) and how these requirements fits into the broader context of particle physics theories. This pursuit is, however, currently marred by limitations in theory predictions of the gravitational wave signal. The dominant complication arising from the need for lattice simulations to compute the signal. Here I will show on-going work and chart the path forward to resolving these limitations.