This paper presents a numerical study of magnetic reconnection processes between magnetically organized structures in the solar wind (e.g., magnetic clouds or plasma bubbles) and the solar wind magnetic field. Magnetic reconnection occurs at points along the surface bounding the two flux systems where antiparallel field lines are forced together. For a magnetic cloud moving outward in an unidirectional radial magnetic field, the reconnection occurs asymmetrically, the location depending on the sense of rotation of the cloud's magnetic field. When a cloud moves in a current sheet, reconnection occurs either symmetrically along both sides of the interface or not at all, depending on the relative orientation of the cloud and solar wind fields. The effectiveness of the reconnection process increases with increasing relative speed between the cloud and the solar wind. When the cloud moves faster (slower) than the solar wind, reconnection takes places at the leading (trailing) edge of the cloud. While important, reconnection does not appear to be able to destroy the overall magnetic cloud structure over distances of 1 AU.