Fixed-point theorems are mathematical principles that assert the existence of points that remain unchanged under specific functions. In simpler terms, if you apply a function to a certain point, that point will map back to itself. These theorems are crucial in various fields, including mathematics, economics, and computer science, as they help in solving equations and optimization problems. One of the most famous fixed-point theorems is the Brouwer Fixed-Point Theorem, which states that any continuous function mapping a compact convex set to itself has at least one fixed point. Another important example is the Banach Fixed-Point Theorem, which provides conditions under which a unique fixed point exists for contraction mappings.