False vacuum decay plays an important role in many branches of physics. In many systems, the initial state is in local thermodynamic equilibrium around the metastable minimum. For such systems the Euclidean path integral is a powerful tool to compute the decay rate or the shape of the true vacuum bubble. On the other hand, the Euclidean approach does not capture real-time dynamics of the phase transition such as the bubble formation and growth, clustering of the bubbles, etc. Furthremore, it is, in general, inapplicable to the decay in out-of-equilibrium states which occur ubiquitously in the universe (preheating, black holes, etc.). Other methods are needed to address these questions and to test the predictions of the Euclidean theory. In the first part of my talk I will review the Euclidean formalism, and in the second part I will discuss numerical real-time simulations as an alternative method to study the decay in the high-temperature (classical) regime. Interestingly, for a thermal initial state the decay rate measured in simulations is lower than predicted by the Euclidean theory. The discrepancy is due to the violation of thermal equilibrium during the bubble nucleation, which happens unavoidably in many commonly studied field theories. This points to the limitation of the Euclidean approach to compute the decay rate even for equilibrium metastable states.
