This thesis studies three problems in online learning. For all the problems the proposed solutions are simple yet non-trivial adaptations of existing online machine learning algorithms. For the task of sequential prediction, a modified multiplicative update algorithm that produces small and accurate models is proposed. This algorithm makes no assumption about the complexity of the source that produces the given sequence. For the task of online learning when examples have varying importances, the proposed algorithm is a version of gradient descent in continuous time. Finally, for the task of efficient online active learning, the implementation we provide makes use of many shortcuts. These include replacing a batch learning algorithm with an online one, as well as a creative use of the aforementioned continuous time gradient descent to compute the desirability of asking for the label of a given example. As this thesis shows, online machine learning algorithms can be easily adapted to many new problems.