Let f be a conformal mapping of a plane domain D onto the unit disk B. The authors are concerned with establishing (uniform) Hölder continuity of f or \(f^{-1}\) on D or B respectively, with Hölder exponents depending on various geometric characteristics of D. Here the geometric notions used are that of b-arcwise connectedness and some variations thereof. A set A in \({\mathbb{C}}\) is b-arcwise connected if each pair of points z and w in A can be joined by an arc \(\gamma\) in A for which \(diam(\gamma)\leq b| z-w|.\) The authors prove, for example, that if D is b-arcwise connected then f is Hölder continuous on D with exponent \(\pi /2(\pi -\arcsin(frac{1}{2}b))\) and if \({\mathbb{C}}-D\) is b- arcwise connected then \(f^{-1}\) is Hölder continuous on B with exponent \(2/(\pi^ 2(1+b)^ 2).\) The proofs involve estimates of harmonic measure and extremal length. Local versions of the results are also established.