Abstract: In this talk we will investigate various ways to quantify the size of self-similar subsets of R and the self-similar measures concentrated on them. Of particular interest will be measures that arise from `overlapping’ iterated function systems, such as Bernoulli convolutions or m-fold convolutions of Cantor measures. These measures have been of interest to mathematicians for more than 80 years and yet fundamental questions remain open and surprising discoveries are still being made.