Twin support vector machine (TSVM) is a very classical and practical classifier for pattern classification. However, the traditional TSVM has two limitations. Firstly, it uses the L_2-norm distance metric that leads to its sensitivity to outliers. Second, it needs to select the appropriate kernel function and the kernel parameters for nonlinear classification. To effectively avoid these two problems, this paper proposes a robust capped L_1-norm kernel-free quadratic surface twin support vector machine (CL_1QTSVM). The strengths of our model are briefly summarized as follows. 1) The robustness of our model is further improved by employing the capped L_1 norm distance metric. 2) Our model is a kernel-free method that avoids the time-consuming process of selecting appropriate kernel functions and kernel parameters. 3) The introduction of L_2-norm regularization term to improve the generalization ability of the model. 4) To efficiently solve the proposed model, an iterative algorithm is developed. 5) The convergence, time complexity and existence of locally optimal solutions of the developed algorithms are further discussed. Numerical experiments on numerous types of datasets validate the classification performance and robustness of the proposed model.