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In the present thesis, the author reviews the physics of cosmological first-order phase transitions that may have occured shortly after the Big Bang. Such transitions proceed via the nucleation and expansion of true vacuum bubbles and give rise to a rich phenomenology, for instance the emission of a stochastic gravitational-wave background caused by bubble collisions. The author discusses, in depth, the formalism of the effective scalar potential and its different contributions in the loop expansion, points out the necessary ingredients for a first-order transition, and assesses the detectability of the associated gravitational-wave spectrum via future space-based observatories and pulsar timing arrays. He then applies the the developed phenomenological toolbox to investigate the detection prospect for phase transitions in the context of specific theories such as the Vev Flip-Flop (a dark matter mechanism) and the Dark Photon Model.