Bounds are determined for the Ramsey number of the union of graphs versus a fixed graph H, based on the Ramsey number of the components versus H. For certain unions of graphs, the exact Ramsey number is determined. From these formulas, some new Ramsey numbers are indicated. In particular, if . Where ki is the number of components of order i and t1 (H) is the minimum order of a color class over all critical colorings of the vertices of H, then .