The Mordell-Weil Theorem is a fundamental result in the field of number theory, specifically concerning elliptic curves. It states that the group of rational points on an elliptic curve over the rational numbers is finitely generated. This means that all rational solutions can be expressed as a finite combination of a finite number of points, along with a torsion subgroup. In simpler terms, if you have an elliptic curve defined by a specific equation, the rational points you can find on that curve can be organized into a manageable structure. This theorem has significant implications in various areas of mathematics, including algebraic geometry and cryptography.