Various growth models for the aggregation of atoms or molecules on a surface shall be discussed in this lecture. A simple random walk simulates the diffusion-limited aggregation. The resulting clusters are fractal structures, which means that they show self-similar behaviour with voids on all length scales. As proposed by Mandelbrot this behaviour can be characterized by a fractal dimension which is smaller than the Euclidean dimension of the surface, i.e., smaller than 2. This fractal dimension can be influenced by appropriate growth conditions, for example weaker fluctuations can be achieved by a noise reduction algorithm leading to more or less dense coverage of the surface. – The singularity spectrum describes the frequency with which the aggregation happens in different regions of the cluster. For diffusion-limited aggregation a universal curve can be obtained which reflects the multifractality of the model. – Several kinetic growth models for the simulation of multi-layered surface growth like the Eden model and its variants shall be analysed. Here the self-affine properties are of interest, because the fractal structures become invariant under anisotropic scale changes. – Different scaling exponents describe the roughness of the surface in dependence on the deposition time and the length scale. As these fluctuations of the width of the surface are strongly correlated over a certain correlation length, it is possible to derive scaling relations between different exponents. These concepts shall also be applied to percolation clusters, where the probability to find an infinite cluster serves as an order parameter describing a continuous phase transition. Different types of percolation like site percolation, bond percolation, as well as continuous percolation models lead to the same critical exponents, which means that these models belong to the same universality class. In this context the renormalization group method shall be introduced, which is a powerful procedure to determine the location of the critical point and the critical exponent.