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This work aims to explore several research directions in stochastic sequential decision-making. The stochastic multi-armed bandit is a theoretical framework in which an agent must repeatedly decide among a predefined set of possible actions while receiving sequential feedback. The main challenge is balancing exploiting the most promising actions with exploring the whole set. The classic model is restrictive when it comes to real-world applications: some of its core assumptions, such as the stationarity of the rewards, the good behavior of the noise, or the absence of constraints, severely limit the breadth of realistic applications for multi-armed bandits. This research investigates the theoretical foundations of such settings, where assumptions are relaxed or completely removed. Some algorithms are proposed to deal with these complex environments, and their theoretical guarantees are characterized.