In this talk, we discuss Lepski's method of adaptive estimation and its applications in different areas such as inverse problems (on imaging) and multi-armed bandits. In the inverse problems setup, we propose a locally adaptive strategy for estimating a function from its Exponential Radon Transform (ERT) data, without any knowledge of the smoothness of functions that are to be estimated. We build a nonparametric estimator and show that our proposed strategy follows the minimax optimal rate up to a log n factor.  We also discuss adaptive estimation in multi-armed bandits based on Qian & Yang, 2016 paper, where they apply Lepski’s method to estimate the smoothness of the mean reward functions. Then, we finally discuss adaptive estimation in the kernel ridge regression setting.