In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ43-model.   In the first half of this talk, we illustrate our main objects and questions through Hamiltonian ODEs, which serve as a toy-model. We also connect our theorem with classical and recent developments in constructive QFT, dispersive PDEs, and stochastic PDEs.   In the second half of this talk, we first discuss the construction and properties of the Gibbs measure. Then, we turn to the most difficult aspect of our argument, which is the probabilistic well-posedness of the cubic nonlinear wave equation. This part combines ingredients from dispersive equations, harmonic analysis, and probability theory.   This is joint work with Y. Deng, A. Nahmod, and H. Yue.