  We study the compactness problem for moduli spaces of holomorphic supercurves
which, being motivated by supergeometry, are perturbed such as to allow for
transversality. We give an explicit construction of limiting objects for
sequences of holomorphic supercurves and prove that, in important cases, every
such sequence has a convergent subsequence provided that a suitable extension
of the classical energy is uniformly bounded. This is a version of Gromov
compactness. Finally, we introduce a topology on the moduli spaces enlarged by
the limiting objects which makes these spaces compact and metrisable.
