A classical theorem of F. and M. Riesz, proved more than 100 years ago, established mutual absolute continuity of harmonic measure and arc length measure, on the boundary of a simply connected domain in the complex plane with a rectifiable boundary. In this talk, we shall discuss recent progress towards understanding the relationship between absolute continuity properties of harmonic measure in a domain Ω ⊂ Rd, and rectifiability of the boundary of Ω.