[en] Various features of the observable universe can be understood as the result of nonequilibrium processes during the early stages of its history, when it was filled with a hot primordial plasma. In many cases, including cosmological freezeout processes, only a few degrees of freedom were out of equilibrium and the background plasma can be viewed as a large heat bath to which these couple. We study scalar and fermionic quantum fields out of thermal equilibrium that are weakly coupled to a large thermal bath with the goal to formulate a full quantum mechanical description of such processes. The bath composition need not be specified. Our analysis is based on Kadanoff-Baym equations, which are the exact equations of motion for the correlation functions in a nonequilibrium quantum system. We solve the equations of motion for the most general Gaussian initial density matrix, without a specific ansatz or a-priori parameterisation and for arbitrarily large deviations from equilibrium. The solutions depend on integral kernels that contain memory effects. These can in good approximation be solved analytically when the field excitations have a small decay width. The full solutions are compared to results obtained by other methods. We prove that the description in terms of a stochastic Langevin equation is equivalent to the Kadanoff-Baym equations. We show the emergence of standard Boltzmann equations as a limit of the Kadanoff-Baym equations in a dilute gas when coherences play no role and discuss quantum Boltzmann equations as an intermediate step. We analyse the properties of the solutions in terms of the equation of state and investigate the validity and implications of quasiparticle approximations. We find that the equation of state can deviate significantly from that of a gas of quasiparticles even if the resonances in the plasma show quasiparticle behaviour in decays and scatterings. A detailed discussion is devoted to the influence of modified dispersion relations and widths in the plasma on gain and loss rates. We illustrate our results in two models for the bath composition, a scalar and a Yukawa model. In both cases we give analytic expressions for the imaginary parts of the self energies, which govern the gain and loss rates. Finally, we discuss applications in cosmology. Our results provide a toolkit for a full quantum mechanical description of cosmological freezeout processes. We discuss the application to thermal leptogenesis, where quantum effects are likely to be of great relevance. The scalar model can also be used to describe the late phase of reheating. In this context, we analyse under which circumstances large thermal masses can put an upper bound on the reheating temperature. (orig.)