In recent years, the connection between the Riesz transform and rectifiability has been essential for the solution of some free boundary problems involving harmonic measure. In my talk I will describe the connection between the Riesz transform and harmonic measure and I will survey some joint results with Damian Dabrowski which characterize the measures $\mu$ with $L^2(\mu)$ bounded Riesz transform in terms of the $\beta_2$ coefficients of the measure, as well as a recent refinement which may have further applications to the study of harmonic measure.