A stochastic gravitational wave background of cosmological origin is an intriguing possibility to be probed by gravitational wave detectors such as pulsar timing arrays and LISA in the near future. In this seminar, I will discuss cosmological phase transitions as a source of GWs and present the novel “Higgsless” simulation approach to predict the stochastic gravitational wave spectrum from such transitions. I will present recent results for strong phase transitions, demonstrate an application to phase transitions seeded by domain walls, and emphasize the critical role of simulations in aligning theoretical predictions with experimental progress.