Abstract: Regression analysis is frequently used in the analyses of spatial data. In this paper, we propose a flexible quantile spatially varying coefficient model to assess how conditional quantiles of the response depend on covariates, allowing the coefficient function to vary with the spatial locations. The model can be used to explore spatial non-stationarity of a regression relationship for heterogeneous spatial data distributed over a domain of a complex or irregular shape. For model estimation, we propose a quantile regression method adopting the bivariate penalized spline technique to approximate the unknown functional-coefficients. Under some regularity conditions, the L2 convergence of the proposed estimator is established with optimal convergence rate. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is developed to solve the optimization problem. Simulation studies have confirmed the excellent performance of the proposed approach. The proposed method is also illustrated by the analysis of mortality data in the U.S.