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Many numerical simulations, in particular that of a two dimensional incompressible free boundary flow, can be done by performing conformal mapping of the flow domain onto a half plane. The detailed behavior of the conformal mapping, which is closely related to the detailed behavior of the solution to a two dimensional Dirichlet problem, is analysed. A uniform asymptotic expansion to the conformal map of a slender domain is constructed. Its salient features are explained and later generalized by theorems valid for arbitrary domains. It is demonstrated that the conformal map onto a disk can not expand distances beyond a certain bound but can be extremely contracting. The logarithm of its derivative is shown to be well behaved. A general perturbation formula from an arbitrary domain to an arbitrary domain which preserves many features of the infinitesmal perturbation formula is derived, and its use is demonstrated on a fractal. These results utilize two estimates, correct up to a constant factor, of the conformal distance and the location of its geodesics.