Abstract: We present a framework for the computation of harmonic and conformal mappings in the plane with mathematical guarantees that the computed mappings are C∞, locally injective and satisfy strict bounds on the conformal and isometric distortion. Such mappings are very desirable in many computer graphics and geometry processing applications. We establish the sufficient and necessary conditions for a harmonic planar mapping to have bounded distortion. Our key observation is that these conditions relate solely to the boundary behavior of the mapping. This leads to an efficient and accurate algorithm that supports handle-based interactive shape-and-image deformation and is demonstrated to outperform other state-of-the-art methods.