  In the group testing problem the aim is to identify a small set of $k\sim
n^\theta$ infected individuals out of a population size $n$, $0<\theta<1$. We
avail ourselves of a test procedure capable of testing groups of individuals,
with the test returning a positive result iff at least one individual in the
group is infected. The aim is to devise a test design with as few tests as
possible so that the set of infected individuals can be identified correctly
with high probability. We establish an explicit sharp
information-theoretic/algorithmic phase transition $\minf$ for non-adaptive
group testing, where all tests are conducted in parallel. Thus, with more than
$\minf$ tests the infected individuals can be identified in polynomial time
\whp, while learning the set of infected individuals is
information-theoretically impossible with fewer tests. In addition, we develop
an optimal adaptive scheme where the tests are conducted in two stages.
