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Introduction Chapter 1.Motivation and statement of the main results 1.Characterization (1)¦Á: Approximation with planes 2.Characterization (2)¦Á: Introducing BMO and VMO 3.Multiplicative vs.additive formulation: Introducing the doubling condition 4.Characterization (1)¦Á and flatness 5.Doubling and asymptotically optimally doubling measures 6.Regularity of a domain and doubling character of its harmonic measure 7.Regularity of a domain and smoothness of its Poisson kernel Chapter 2.The relation between potential theory and geometry for planar domains 1.Smooth domains 2.Non smooth domains 3.Preliminaries to the proofs of Theorems 2.7 and 2.8 4.Proof of Theorem 2.7 5.Proof of Theorem 2.8 6.Notes Chapter 3.Preliminary results in potential theory 1.Potential theory in NTA domains 2.A brief review of the real variable theory of weights 3.The spaces BMO and VMO 4.Potential theory in C1 domains 5.Notes Chapter 4.Reifenberg flat and chord arc domains 1.Geometry of Reifenberg flat domains 2.Small constant chord arc domains 3.Notes Chapter 5.Further results on Reifenberg fiat and chord arc domains 1.Improved boundary regularity for J-Reifenberg flat domains 2.Approximation and Rellich identity 3.Notes Chapter 6.From the geometry of a domain to its potential theory 1.Potential theory for Reifenberg domains with vanishing constant 2.Potential theory for vanishing chord arc domains 3.Notes Chapter 7.From potential theory to the geometry of a domain 1.Asymptotically optimally doubling implies Reifenberg vanishing 2.Back to chord arc domains 3.log k E VMO implies vanishing chord arc; Step I 4.log k E VMO implies vanishing chord arc; Step II 5.Notes Chapter 8.Higher codimension and further regularity results 1.Notes Bibliography