### Abstract:

In this work the growth instabilities of vicinal crystal surfaces during MBE are studied theoretically. The vicinal surfaces grow in a step-flow mode, where steps present due to a small miscut relative to a high symmetry direction of a crystal, propagate due to a deposition flux. The number of the steps remains constant as nucleation of atomic island is suppressed by the presence of steps. In the first part the models used to describe the step-flow growth are presented. Also the different microscopic processes taking place on a growing surfaces are discussed at length, as the large scale morphology is determined by the relative relevance of these processes. The dynamics of atoms diffusing along the atomic steps, which are of central importance for the step-flow growth, are addressed in particular. The main subject of this thesis are the step meandering instabilities, which lead to a ripple morphology on a growing surface. The atomic steps become wavy due to growth instabilities as they propagate. The wave patterns formed on the steps are in-phase over multiple steps, thus leading to long ripples running in the direction of the step-train. The wavelength of the pattern, i.e. the typical separation of the ripples, is set by the competition between the driving force (the deposition flux) and the relaxation of the steps. In equilibrium the steps are straight. In growth experiments the typical scale of the ripples lies in the nanometer scale l ~ 10-1000 nm. The dynamics of these instabilities are studied employing Monte-Carlo simulations and partial differential equations, describing the time evolution of the steps. A quantitative comparison between these two approaches is made. The results are also related to recent experimental results. In the last part the destabilization of the step-flow growth due to the appearance of new steps is considered. New steps may result from either island nucleation on terraces, or due to the appearance of vacancy islands that are formed when a strongly deformed step crosses itself.