If every vertex in a graph G has the same degree, then the graph is called a regular graph. That is, if deg(v) = r for all vertices in the graph, then it is denoted as an r-regular graph. A graph G is said to be vertex-weighted if all of the vertices are assigned weights. A generalized definition for degree regularity for vertex-weighted graphs can be stated as follows: A vertex-weighted graph is said to be rw-regular if the sum of the weights in the neighborhood of every vertex is rw. If all vertices are assigned the unit weight of 1, then this is equivalent to the definition for r-regular graphs. In this thesis, we determine if a graph has a weighting scheme that makes it a weighted regular graph or prove no such scheme exists for a number of special classes of graphs such as paths, stars, caterpillars, spiders and wheels.