In this talk I will consider self-affine sets $K\subset \R^n$ arising as attractors of iterated function systems of the form $$ \{A+d_1,\ldots,A+d_k\}, $$ where $d_1,\ldots,d_k\in\R^n$, and $A$ is an affine map. Unlike the case when $A$ is a similarity, determining the dimensional and topological properties of $K$ proves a challenging problem, even when there is complete separation. Some of the natural questions are: what are the Hausdorff and box dimensions of $K$? Is $K$ connected, and if this is the case, does it have nonempty interior? What is the multifractal structure of the natural measures supported on $K$? I will review some of the known partial answers to these questions, and describe some related work in progress. I will focus mainly on planar sets defined by 2 vectors; in this setting the results are rather complete, at least in an ``almost every'' sense. The main tool is the technique developed to prove absolute continuity of Bernoulli convolutions.