  We study the structure of inverse primitive feebly compact semitopological
and topological semigroups. We find conditions when the maximal subgroup of an
inverse primitive feebly compact semitopological semigroup $S$ is a closed
subset of $S$ and describe the topological structure of such semiregular
semitopological semigroups. Later we describe the structure of feebly compact
topological Brandt $\lambda^0$-extensions of topological semigroups and
semiregular (quasi-regular) primitive inverse topological semigroups. In
particular we show that inversion in a quasi-regular primitive inverse feebly
compact topological semigroup is continuous. Also an analogue of Comfort--Ross
Theorem is proved for such semigroups: a Tychonoff product of an arbitrary
family of primitive inverse semiregular feebly compact semitopological
semigroups with closed maximal subgroups is feebly compact. We describe the
structure of the Stone-\v{C}ech compactification of a Hausdorff primitive
inverse countably compact semitopological semigroup $S$ such that every maximal
subgroup of $S$ is a topological group.
