  The main goal of this work is to construct and study a reasonable
compactification of the strata of the moduli space of Abelian differentials.
This allows us to compute the Kodaira dimension of some strata of the moduli
space of Abelian differentials. The main ingredients to study the
compactifications of the strata are a version of the plumbing cylinder
construction for differential forms and an extension of the parity of the
connected components of the strata to the differentials on curves of compact
type. We study in detail the compactifications of the hyperelliptic minimal
strata and of the odd minimal stratum in genus three.
