Many real-world decision-making settings involve uncertainty about the underlying input, future outcomes, or both. In such environments, additional information can often be ac-

quired, but only at a cost. This leads to a fundamental algorithmic question: how should costly information acquisition be incorporated into the decision-making process? These questions were first formalized in Weitzman’s classical Pandora’s Box problem, in which a decision-maker must choose among stochastic alternatives whose realized values may be revealed by paying known inspection costs. The objective is to maximize utility, defined as the value of the final selection minus the total cost of exploration. Since then, this framework has evolved substantially to capture richer and more realistic settings, including combinatorial constraints, multi-stage inspection processes, and multiple modes of information acquisition. In recent years, there has been significant progress in our understanding of costly-information models for online and sequential selection. A growing body of work has developed new algorithmic techniques, uncovered surprising structural connections to other central areas in economics and computer science, and opened a range of compelling new research directions. The goal of this tutorial is to present these developments in a unified and accessible way, introducing the core models, highlighting major recent advances, and emphasizing the many important open questions that remain.