Euclidean quantum field theories can be viewed as probability measures on spaces of field configurations. Stochastic quantization provides a dynamical approach to such theories: one constructs a stochastic evolution whose equilibrium distribution is the measure of interest. I will begin by introducing this idea using the \Phi^4 model as a basic example. Its stochastic quantization leads to a nonlinear stochastic heat equation driven by spacetime white noise. In dimensions two and higher, the noise is so irregular that solutions are distributions rather than ordinary functions, so the cubic nonlinearity is not directly well defined. In dimensions two and three, however, the equation can be given a rigorous meaning after a suitable renormalization, for instance using the theory of regularity structures. On a compact domain, the resulting dynamics has a unique equilibrium and solutions converge to it as time tends to infinity. The situation in infinite volume is considerably more delicate. In particular, uniqueness cannot hold for all parameter values, since at low temperature the corresponding field theory exhibits phase coexistence. I will discuss a result showing that, in dimensions two and three, when the mass is sufficiently large or the coupling constant sufficiently small, all solutions in infinite volume converge exponentially fast to a unique stationary solution, with estimates uniform over the initial condition. As a consequence, the corresponding infinite-volume Φ⁴ measure is unique and its spatial correlations decay exponentially with distance.