Brouwer's Fixed-Point Theorem states that any continuous function mapping a compact convex set to itself has at least one fixed point. This means that if you take a shape like a disk or a square and continuously move points within it, there will always be at least one point that remains in the same position after the transformation. The theorem is named after the Dutch mathematician Luitzen Egbertus Jan Brouwer, who introduced it in the early 20th century. It has important implications in various fields, including topology, economics, and game theory, as it helps to demonstrate the existence of equilibrium points in different systems.