We present two complementary methods for automatically improving mesh parameterizations and demonstrate that they provide a very desirable combination of efficiency and quality. First, we describe a new iterative method for constructing quasi-conformal parameterizations with free boundaries. We formulate the problem as fitting the coordinate gradients to two guidance vector fields of equal magnitude that are everywhere orthogonal. In only one linear step, our method efficiently generates parameterizations with natural boundaries from those with convex boundaries. If repeated until convergence, it produces the unique global minimizer of the Dirichlet energy. Next, we introduce a new non-linear optimization framework that can rapidly reduce interior distortion under a variety of metrics. By iteratively solving linear systems, our algorithm converges to a high quality, low distortion parameterization in very few iterations. The two components of our system are effective both in combination or when used independently.