  We define and investigate the property of being `exponent-critical' for a
finite group. A finite group is said to be exponent-critical if its exponent is
not the least common multiple of the exponents of its proper non-abelian
subgroups. We explore properties of exponent-critical groups and give a
characterization of such groups. This characterization generalises a classical
result of Miller and Moreno on minimal non-abelian groups; interesting families
of $p$-groups appear.
