Crystal growth phenomena are discussed with special reference to growth from vapour. The basic concepts of crystal growth are recalled, including the different growth modes, the dependence of the growth rate on disequilibrium and temperature, and the atomic processes relevant for growth. The methods used in crystal growth simulations are reviewed, with special reference to kinetic Monte Carlo methods. The roughness of growing surfaces, and the roughness properties of the discrete and continuum growth models (the latter being described via stochastic differential equations) are discussed, together with the special phenomena occurring in the vicinity of the roughening temperature. A number of simulations based on the six-vertex model and on kinetic counterparts of the BCSOS model are reviewed. Finally, the instabilities arising during growth are considered, including a discussion of phenomena such as dendritic growth and ramified cluster growth and reviewing the recent, extensive studies concerning unstable MBE growth.

Kinetic roughening in two variants of a solid-on-solid model of epitaxial growth, a ''toy'' relaxation model and a collective diffusion model, is studied and compared, using extensive computer simulations in different spatial dimensions. We have studied the Wolf-Villain ͑WV͒ model and its modifications, and a full diffusion ͑FD͒ model with Arrhenius dynamics. The WV model shows nonuniversal features and its asymptotic behavior switches between the Edwards-Wilkinson type and a morphological instability, depending on the spatial dimension, a lattice coordination, or a minor modification of the model rules. The results for the FD model in 1ϩ1 and 2ϩ1 dimensions are not consistent with any of the continuum equations proposed to describe epitaxial growth; in particular, we observe too low values of the roughness exponent measured from the height-difference correlation function. In 1ϩ1 dimensions, the results obtained for both models are very similar for more than 10 6 monolayers deposited with an ''incorporation'' radius in the WV model corresponding to the substrate temperature in the FD model. Such a close correspondence is not found in 2ϩ1 dimensions. Asymptotic behavior of the WV and FD models is different in all spatial dimensions. Both FD and WV models show anomalous scaling in all dimensions studied. The anomalous scaling in the FD model is very weak ͑logarith-mic͒ in the physically relevant case of 2ϩ1 dimensions.

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