The Phi^4, and generally Phi^p measures, which are extensively studied in quantum field theory, also occur naturally as invariant Gibbs measures for certain (dispersive) Hamiltonian PDEs and parabolic SPDEs. A fundamental question is to rigorously justify the invariance of such measures under said dynamics, which leads to deep questions in the solution theory of random data and stochastic PDEs. In this talk we review some recent progress in the dispersive setting, including the proof of invariance of Phi_2^p under Schrodinger dynamics and of Phi_3^4 under wave dynamics. In the Schrodinger case, we also obtain local well-posedness results in the full probabilistically subcritical regime. These are joint works with Bjoern Bringmann, Andrea R. Nahmod and Haitian Yue.