  We gather conditions on a class H of continuous maps of topological spaces
that allow a reasonable theory of fibrations up to an equivalence (a map from
this class) which we call H-fibrations. The weak homotopy equivalences recover
quasifibrations and homology equivalences yield homology fibrations. We study
local H-fibrations that behave nicely with respect to homotopy colimits
together with universal H-fibrations that behave nicely with respect to
pullbacks. We then proceed to classify H-fibrations up to a natural notion of
equivalence.
