The purpose of this research is to investigate robust loss functions and robust boosting algorithms when there are outliers in the training data. Boosting methods, which are algorithms included loss function and weak classifiers, are a way of predicting class label of given inputs. Loss functions can be useful in weak classifiers for boosting algorithms, and it is the way of penalizing misclassification. In addition, robust loss function gives penalties not only misclassification but also true classification margin in order to get more stable classifiers. In this paper, a new robust boosting algorithm, GudermannianBoost, is proposed. In the application part, there is a simulation study between two boosting algorithms, which have similar loss functions. Moreover, there are some real datasets applications in the presence of outlier-ridden datasets. All results from the simulation and application highlight the importance of robust algorithms for classifying observations more accurately.