Among recent accomplishments in the field of the transuranium elements have been the discoveries of elements 99–102. The fact that these elements are members of the actinide transition series and have chemical homologues in the lanthanide or rare-earth series of elements has provided a basis for their initial identification. Special techniques have also been required, however, for their discovery. In the case of element 101, when only one or two atoms per bombardment were synthesized, a new method, the recoil technique, was used to separate the product from the target material by purely physical means. Element 102 is the first element to be discovered as a product of heavy-ion bombardment. This element, which has a half-life of approximately three seconds, was identified chemically by means of its daughter Fm250. The element 102 and Fm250 atoms were isolated by an ingenious adaptation of the recoil technique which was used in the discovery of element 101. Indications are that new transuranium elements will be discovered, but research along this line is extremely complex and difficult. One of the problems to be solved is concerned with the availability of target materials of high atomic number. These are synthesized by the long-term neutron irradiation of plutonium. One such irradiation program has supplied us with californium and berkelium in macroscopic amount. As an interesting result of this program the first pure californium compounds have been prepared, and studies of their properties are in progress. A national program for the production of heavy isotopes is expected to yield milligram amounts of californium by 1965. The use of heavy-ion bombardments offers the most promise for the synthesis of new elements, and work on the preparation of element 103 and heavier elements by this means is in progress. Fission predominates in such nuclear reactions, and thus only extremely small yields of elements of high atomic number can be obtained. Another difficulty lies in the fact that the elements beyond element 102 are expected to have very short half lives. These difficulties indicate that new methods for their identification need to be used. Although the position of these new elements in the periodic table can be predicted so that their chemical nature can be anticipated, the first identifications will probably not be made by traditional chemical methods.