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In this paper, we study a classification problem in which sample labels are randomly corrupted. In this scenario, there is an unobservable sample with noise-free labels. However, before being observed, the true labels are independently flipped with a probability $\rho \in [0,0.5)$ , and the random label noise can be class-conditional. Here, we address two fundamental problems raised by this scenario. The first is how to best use the abundant surrogate loss functions designed for the traditional classification problem when there is label noise. We prove that any surrogate loss function can be used for classification with noisy labels by using importance reweighting, with consistency assurance that the label noise does not ultimately hinder the search for the optimal classifier of the noise-free sample. The other is the open problem of how to obtain the noise rate \rho$ . We show that the rate is upper bounded by the conditional probability P(\hat{Y}|X) of the noisy sample. Consequently, the rate can be estimated, because the upper bound can be easily reached in classification problems. Experimental results on synthetic and real datasets confirm the efficiency of our methods. [ABSTRACT FROM PUBLISHER]