The cost of an accelerator depends strongly on the requirements that it must satisfy to drive a target. Therefore, an important part of the Heavy-Ion Fusion Systems Assessment (HIFSA) Project has been a search for, and an assessment of, target concepts that might relax the accelerator requirements. This paper outlines the considerations that have guided the search for improved targets and gives a brief description of the various concepts that have been studied. Not all of the target concepts were sufficiently developed for inclusion in the HIFSA study and are discussed here for completeness. Recent work has led to new estimates of the gain of radiatively driven targets. This work was not completed in time for the HIFSA study, but is included in this paper. Although the new results differ substantially from the base case used in the study, a systems study performed at Lawrence Livermore National Laboratory shows that the new results increase the cost of electricity by slightly less than 10%.