  We develop a pseudodifferential calculus for differential operators
associated to quasi-fibered boundary metrics (QFB metrics), a class of metrics
including the quasi-asymptotically conical metrics (QAC metrics) of
Degeratu-Mazzeo and the quasi-asymptotically locally Euclidean metrics (QALE
metrics) of Joyce. Introducing various principal symbols, we introduce the
notion of fully elliptic QFB operators and show that those are Fredholm when
acting on suitable Sobolev spaces. For QAC metrics, we also develop a
pseudodifferential calculus for the conformally related class of Qb metrics. We
use these calculi to construct a parametrix for the Hodge-deRham operator of
certain QFB metrics, allowing us to obtain results about the decay at infinity
of $L^2$ harmonic forms. In a companion paper, this result will be used to
study the $L^2$ cohomology of some QFB metrics. For QFB metrics of depth two,
our parametrix construction allows us to show that the Hodge-deRham operator is
Fredholm when acting on suitable Sobolev spaces.
