The lifetime of layer-by-layer growth of crystal surfaces, mainly in the context of growth conditions found in molecular beam epitaxy (MBE), is the central issue of this thesis. These conditions imply a driven system far from equilibrium which relaxes due to surface diffusion. At first, the ceasing of layer-by-layer growth due to fluctuations in the particle supply is considered. A theory for the according lifetime is presented and confirmed for the one-dimensional surface. Special care is taken for the two-dimensional case where deviations from previous results are found, explained, and used to revise the assumptions on which the theory is based. In particular the applicability of the -- commonly accepted -- conserved KPZ continuum equation and the premise of a single morphologically relevant length scale are affected. The practically more relevant scenario of layer-by-layer growth's breakdown caused by barriers to interlayer transport (which give rise to the Villain instability) is studied. Data obtained fr om computer simulations is compared to the predictions of a linear stability analysis and is used to foretell the effect of counteracting variations of energy barriers. The latter enables to decide in which cases a strained surface is either hindering or advantageous for layer-by-layer growth. A mean field model describing surface growth, which lacked up to now a systematic treatment, is investigated. For the basic version, the asymptotic behavior is derived exactly and -- tuning the sole control parameter -- a transition from Poisson-like growth to persistent layer-by-layer growth is found together with a non-trivial powerlaw behavior right at the transition point. Finally the extensibility of the model to include a finite lifetime of layer-wise growth is examined. The damping of oscillations of certain surface-sensitive quantities is the manifestation of the surface's roughening which terminates the layer-by-layer growth. A scenario alternative to the roughening is suggested. It leads as well to damping of oscillations and consists of a step bunch which dissolves during growth and 'floods' an adjacent terrace. Growth simulations of this process are compared to a deterministic model and to experimental results. Finally several toy models for surface growth, subjected to noise reduction are considered. The latter technique makes possible layer-by-layer growth also in these models and the dependence of its lifetime on the degree of the noise reduction is studied. The main focus is on the behavior's relation to continuum equations and the corresponding universality classes, which are commonly used to classify the different models.