  We systematically develop Bridgeland's and Bridgeland-Maciocia's techniques
for studying elliptic fibrations, and identify criteria that ensure 2-term
complexes are mapped to torsion-free sheaves under a Fourier-Mukai transform.
As an application, we construct an open immersion from a moduli of stable
complexes to a moduli of Gieseker stable sheaves on elliptic threefolds. As
another application, we give various 1-1 correspondences between fiberwise
semistable torsion-free sheaves and codimension-1 sheaves on Weierstrass
surfaces.
