Group testing is a basic paradigm for experimental design. In computational biology, group testing problems come up in designing experiments with sequences for mapping, screening libraries, etc. While a great deal of classical research has been done on group testing over the last fifty years, the current biological applications bring up many new issues in group testing which had not been previously considered. The authors isolate the most basic issues in molecular biological group testing. Given these, they formulate a set of novel group testing problems for designing cost effective experiments. For some of these problems they give solutions, while leaving others open.