We consider calculation of the dimensions of self-affine fractals and multifractals that are the attractors of iterated function systems specified in terms of upper triangular matrices. Using methods from linear algebra we obtain explicit formulae for the dimensions that are valid in many cases.

We consider several different models for generating random fractals including random self-similar sets, random self-affine carpets, and fractal percolation. In each setting we compute either the almost sure or the Baire typical Assouad dimension and consider some illustrative examples. Our results reveal a common phenomenon in all of our models: the Assouad dimension of a randomly generated fractal is generically as big as possible and does not depend on the measure theoretic or topological structure of the sample space. This is in stark contrast to the other commonly studied notions of dimension like the Hausdorff or packing dimension.Mathematics Subject Classification 2010: primary: 28A80, 60J80; secondary: 37C45, 54E52, 82B43.

We study the gap sequence of totally disconnected McMullen sets. Our result shows that if every horizontal line in the McMullen set is nonempty, then the gap sequence is unrelated to the box dimension. This implies that in such situations, the separation properties of McMullen sets are quite different from that of self-similar sets.

Under certain conditions the 'singular value function' formula gives the Hausdorff dimension of self-affine fractals for almost all parameters in a family. We show that the size of the set of exceptional parameters is small both in the sense of Hausdorff dimension and Fourier dimension.

We find the almost sure Hausdorff and box-counting dimensions of random subsets of self-affine fractals obtained by selecting subsets at each stage of the hierarchical construction in a statistically self-similar manner.