Abstract. We discuss some results and open questions in the field of classical self-similar constructions: Boundaries of self-similar sets with open set condition; the dimension of a self-similar set with a big overlapping; the singularity of self-similar measures with respect to Hausdorff and packing measures and a variational property of self-similar measures which plays a role in multifractality. The multidimensional Legendre Transform is shown to satisfy the multifractal formalism for intersections of several multifractal layers relative to different self-similar measures.