  In this paper we consider heterotic compactifications on K3 x T2 as well as
type II compactifications on K3-fibred Calabi-Yau spaces with certain fluxes
for the gauge and RR field strengths F and H turned on. By providing an
identification of corresponding fluxes we show that the well-known N=2
heterotic/type II string-string duality still holds for a subset of all
possible fluxes, namely those which arise from six-dimensional gauge fields
with internal magnetic flux on the common two-sphere P1, which is the base
space of the type II K3-fibration. On the other hand, F- and H-fluxes without
P1-support, such as heterotic F-fluxes on the torus T2 or type II H-fluxes on
cycles of the K3-fibre cannot be matched in any simple way, which is a
challenge for heterotic/type II string-string duality. Our analysis is based on
the comparison of terms in the effective low-energy heterotic and type II
actions which are induced by the fluxes, such as the Green-Schwarz couplings
related to flux-induced U(1) anomalies, the effective superpotential and the
Fayet-Iliopoulos scalar potential.
