Both cost-sensitive classification and online learning have been studied extensively in data mining and machine learning communities, respectively. It is a bit surprising that there was very limited comprehensive study for addressing an important intersecting problem, that is, cost-sensitive online classification. In this paper, we formally study this problem, and propose a new framework for cost-sensitive online classification by exploiting the idea of online gradient descent techniques. Based on the framework, we propose a family of cost-sensitive online classification algorithms, which are designed to directly optimize two well-known cost-sensitive measures: (i) maximization of weighted sum of sensitivity and specificity, and (ii) minimization of weighted misclassification cost. We analyze the theoretical bounds of the cost-sensitive measures made by the proposed algorithms, and extensively examine their empirical performance on a variety of cost-sensitive online classification tasks.