Moment problems are mathematical challenges that involve determining a probability distribution based on its moments, which are specific numerical values that summarize certain characteristics of the distribution. The moments can include measures like the mean, variance, and higher-order moments, and they provide insights into the shape and behavior of the distribution. There are two main types of moment problems: the Hamburger moment problem, which deals with distributions on the entire real line, and the Stieltjes moment problem, which focuses on distributions defined on a semi-infinite interval. Solving these problems can help in various fields, including statistics and physics, by revealing underlying patterns in data.