Large time behaviour of reaction-diffusion systems is investigated in two aspects: convergence to equilibrium and random attractors. In the first part, by exploiting the so-called entropy method, exponential convergence to equilibrium for a large class of systems arising from chemical and biochemical reaction networks, partially without detailed balance equilibria, is shown with computable rates. A quasi-steady-state approximation and fast reaction limit for reactive systems are also studied. The second part deals with random attractors for stochastic equations on unbounded domains. The existence and upper semicontinuity of random attractors is investigated for a Navier-Stokes-Voigt equation and the regularity of random attractors is studied for a nonlinear reaction-diffusion equation and a FitzHugh-Nagumo system.