We present LSCM (Least Squares Conformal Maps), a new quasi-conformal parameterization method for triangulated surfaces, based on a least-squares approximation of the Cauchy-Riemann equations. We show the following properties: Our criterion minimizes angle deformations and non-uniform scaling. The objective function is a quadratic form, that can be efficiently minimized. We prove that this quadratic form is definite positive, therefore the existence and uniqueness of the minimum is ensured. The borders of the charts do not need to be fixed. Therefore, large charts with arbitrarily shaped borders can be parameterized.