Along with the rapid growth in the size of data generated and collected over time, the need for developing online algorithms that can provide answers without any offline training has considerably increased. In this thesis, we consider the prediction with expert advice problem under the online learning framework. Specifically, we consider problems where experts have asymmetric information about the sample space. First, we propose an algorithm that selects a subset of the experts and makes predictions based on the advices of this subset. Then, we propose another algorithm that clusters samples in an online manner and makes predictions based on the history of observations and decisions within each cluster. Next, we consider the Safe Bandit, a variant of the Risk Aware Multi Armed Bandit, where the goal is to minimize the number of rounds in which a risky arm is chosen. Adopting mean-variance as the risk notion, we define an arm as risky if its mean-variance is higher than a given threshold. Using this, we define a new regret measure called Risk Violation Regret (RVR), which depends on the number of times risky arms are selected. Then, we propose a learning algorithm called Exploration and Exploitation with Risk Thresholds (EXERT), and prove that it achieves O(1) RVR with high probability. Afterwards, we use EXERT in an expert selection problem, where each expert corresponds to a neural network with reject option. For this, we propose a method to train these neural networks and use them to evaluate the performance of EXERT in real-world datasets.