This series of talks is an extension of the colloquium talk where we outline a proof of recent results concerning the structure of stationary and minimizing harmonic maps.  Specifically, we will show that the singular stratum S^k(f) are k-rectifiable for a stationary map, and for a minimizing map that the singular set S(f) has finite n-3 measure.  We will also show sharp sobolev estimates for solutions.  The first lecture will focus on a new series of Reifenberg-type results.  Specifically, we give criteria under which a set in R^n can be shown to be rectifiable with uniform k-dim hausdorff measure estimates.  Such a result is only useful of course if we can apply it to a concrete situation, and in order to apply this to the singular stratum S^k(f) we prove a new L^2 subspace approximation theorem.  In the second lecture we will discuss this in detail, as well as the quantitative stratification, which allows us to decompose S^k into more controllable pieces for the analysis.  This is joint work with Daniele Valtorta.