False vacuum decay plays a pivotal role in the physics of the early Universe. However, we lack a satisfying theoretical understanding of this process, with existing approaches working only in imaginary (Euclidean) time, and relying on untested assumptions and approximations. In this talk, I will introduce two complementary approaches toward a real-time theory of vacuum decay: semiclassical numerical simulations on the lattice, and quantum analogue experiments in the laboratory. I will describe some of the new physical insights emerging from these approaches, and their implications for inflation, baryogenesis, and gravitational-wave cosmology.