This paper introduces a novel Robust Regression (RR) model, named Sinkhorn regression, which imposes Sinkhorn distances on both loss function and regularization. Traditional RR methods target at searching for an element-wise loss function (e.g., Lp-norm) to characterize the errors such that outlying data have a relatively smaller influence on the regression estimator. Due to the neglect of the geometric information, they often lead to the suboptimal results in the practical applications. To address this problem, we use a cross-bin distance function, i.e., Sinkhorn distances, to capture the geometric knowledge of real data. Sinkhorn distances is invariant in movement, rotation and zoom. Thus, our method is more robust to variations of data than traditional regression models. Meanwhile, we leverage Kullback-Leibler divergence to relax the proposed model with marginal constraints into its unbalanced formulation to adapt more types of features. In addition, we propose an efficient algorithm to solve the relaxed model and establish its complete statistical guarantees under mild conditions. Experiments on the five publicly available microarray data sets and one mass spectrometry data set demonstrate the effectiveness and robustness of our method.