Image classification systems recently made a giant leap with the advancement of deep neural networks. However, these systems require an excessive amount of labeled data to be adequately trained. Gathering a correctly annotated dataset is not always feasible due to several factors, such as the expensiveness of the labeling process or difficulty of correctly classifying data, even for the experts. Because of these practical challenges, label noise is a common problem in real-world datasets, and numerous methods to train deep neural networks with label noise are proposed in the literature. Although deep neural networks are known to be relatively robust to label noise, their tendency to overfit data makes them vulnerable to memorizing even random noise. Therefore, it is crucial to consider the existence of label noise and develop counter algorithms to fade away its adverse effects to train deep neural networks efficiently. Even though an extensive survey of machine learning techniques under label noise exists, the literature lacks a comprehensive survey of methodologies centered explicitly around deep learning in the presence of noisy labels. This paper aims to present these algorithms while categorizing them into one of the two subgroups: noise model based and noise model free methods. Algorithms in the first group aim to estimate the noise structure and use this information to avoid the adverse effects of noisy labels. Differently, methods in the second group try to come up with inherently noise robust algorithms by using approaches like robust losses, regularizers or other learning paradigms.