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First-order phase transitions play an important role in many areas of physics, from cosmology to condensed matter and atomic physics. Such transitions proceed through the nucleation and growth of bubbles of a stable phase within a metastable one. I will discuss recent advances in understanding the real-time dynamics and non-equilibrium effects accompanying bubble nucleation. These effects arise from departures from local thermal equilibrium and influence the properties of the transition, including the nucleation rate. Local equilibrium may be absent in the initial state of the system or break down during its evolution. I will review methods for calculating the nucleation rate that go beyond the standard equilibrium statistical approach and can account for non-equilibrium effects. I will also present results from real-time classical lattice simulations, which reveal the importance of non-perturbative contributions to the nucleation rate arising from large, coherent fluctuations preceding bubble formation. Finally, I will discuss the implications of these findings for cosmological phase transitions and studies of false vacuum decay in quantum simulators.