The usual goal of online learning is to minimize the regret, which measures the performance of online learner against a fixed comparator. However, it is not suitable for changing environments in which the best decision may change over time. To address this limitation, new performance measures, including dynamic regret and adaptive regret have been proposed to guide the design of online algorithms. In dynamic regret, the learner is compared with a sequence of comparators, and in adaptive regret, the learner is required to minimize the regret over every interval. In this paper, we will review the recent developments in this area, and highlight our contributions. Specifically, we have proposed novel algorithms to minimize the dynamic regret and adaptive regret, and investigated the relationship between them.