In this work, infinitely many families of rational minimal surfaces in Euclidean 3-space are constructed. A rational minimal surface is a complete conformal non-planar minimal immersion whose Weierstrass representation is defined on the Riemann 2-sphere punctured at finitely many points (the puncture points are mapped by the Weierstrass representation to the ends of the surface), and which has the following properties: (1) there are no umbilic points on the surface, (2) the surface has finite total curvature, and hence both the Hopf differential and Gauss map extend meromorphically to the full Riemann 2-sphere, (3) the null curve in \(C^3\) that results from integration in the Weierstrass representation (before taking the real part to produce the minimal immersion in \( R^3)\) is well defined on the same domain as that of the minimal immersion (4) the Hopf differential is non-zero and holomorphic at all but one of the ends. The author shows that under these conditions, one may assume without loss of generality that the Hopf differential equals \( dz^2 \) on the Riemann 2-sphere, and that the Weierstrass representation is then determined solely by the Gauss map, which the author calls the umbilic free representation. The author then gives a method for creating many rational minimal surfaces from a given rational minimal surface (such as Enneper's surface) with Gauss map \(g\), and this method is referred to as the UP-iteration. This method applies a Möbius transformation and then a Darboux-Bäcklund transformation to \(g\) to produce a new Gauss map. However, this new Gauss map is not generally well defined on the Riemann 2-sphere and generally will not produce another rational minimal surface via the umbilic free representation form of the Weierstrass representation. The author finds conditions so that these problems will not occur: require that \(g\) has even-order branch points and that its Schwarzian has only zero residues. With this added requirement, the author shows that the UP-iteration can then be applied not only once, but infinitely many times, producing an infinite sequence of rational minimal surfaces. In the latter half of this work, the author describes examples of rational minimal surfaces produced by this method, and then explains the connection of this work to solutions of the KdV equation.