Cost-sensitive multiclass classification has recently acquired significance in several applications, through the introduction of multiclass datasets with well-defined misclassification costs. The design of classification algorithms for this setting is considered. It is argued that the unreliable performance of current algorithms is due to the inability of the underlying loss functions to enforce a certain fundamental underlying property. This property, denoted guess-aversion, is that the loss should encourage correct classifications over the arbitrary guessing that ensues when all classes are equally scored by the classifier. While guess-aversion holds trivially for binary classification, this is not true in the multiclass setting. A new family of cost-sensitive guess-averse loss functions is derived, and used to design new cost-sensitive multiclass boosting algorithms, denoted GEL- and GLL-MCBoost. Extensive experiments demonstrate (1) the importance of guess-aversion and (2) that the GLL loss function outperforms other loss functions for multiclass boosting.