  We consider topological twists of four-dimensional $\mathcal{N}=2$
supersymmetric QCD with gauge group SU(2) and $N_f\leq 3$ fundamental
hypermultiplets. The twists are labelled by a choice of background fluxes for
the flavour group, which provides an infinite family of topological partition
functions. In this Part I, we demonstrate that in the presence of such fluxes
the theories can be formulated for arbitrary gauge bundles on a compact
four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets,
which introduce new intricacies for the evaluation of the low-energy path
integral on the Coulomb branch. We develop techniques for the evaluation of
these path integrals. In the forthcoming Part II, we will deal with the
explicit evaluation.
