In this talk I will discuss the dynamic critical behavior of relativistic scalar field theories, with dissipative (Model A), diffusive (Model B) and conservative (Model C/D) dynamics. Based on first principle simulations, I will discuss the manifestations of the critical behavior in real-time correlation functions and extract universal scaling functions that describe the spectral functions of the order parameter in the vicinity of the critical point. If time remains I will also present preliminary results for the evolution of cumulants of the order parameter in non-equilibrium phase transitions, where a system dynamically transits the critical point in the phase diagram.