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Uniform rectifiability is a cornerstone of quantitative geometric measure theory in Euclidean spaces that has revealed connections between Harmonic Analysis, PDE, and GMT. However, significant progress in understanding the structure of general uniformly rectifiable (UR) metric spaces has only been achieved in the past few years. In this talk, I will give a brief introduction to uniform (and non-uniform) rectifiability. Then, I will present some new results on characterizations of UR metric spaces involving two quantitative regularity properties for Hausdorff measure: David and Semmes's weak constant density condition (WCD) and control over a metric space variant of Tolsa's alpha number, a quantity that measures local mass transport cost from Hausdorff measure to a flat measure.