  We consider mirror symmetry for (essentially arbitrary) hypersurfaces in
(possibly noncompact) toric varieties from the perspective of the
Strominger-Yau-Zaslow (SYZ) conjecture. Given a hypersurface $H$ in a toric
variety $V$ we construct a Landau-Ginzburg model which is SYZ mirror to the
blowup of $V\times\mathbb{C}$ along $H\times 0$, under a positivity assumption.
This construction also yields SYZ mirrors to affine conic bundles, as well as a
Landau-Ginzburg model which can be naturally viewed as a mirror to $H$. The
main applications concern affine hypersurfaces of general type, for which our
results provide a geometric basis for various mirror symmetry statements that
appear in the recent literature. We also obtain analogous results for complete
intersections.
