We will present joint work with Mohammed Abouzaid and Ludmil Katzarkov, investigating mirror symmetry for blowups from the perspective of the Strominger-Yau-Zaslow conjecture. Namely, we first describe how to construct a Lagrangian torus fibration on the blowup of a toric variety along a codimension 2 subvariety contained in a toric hypersurface. Then we discuss the SYZ mirror and its instanton corrections, to provide an explicit description of the mirror Landau-Ginzburg model (up to higher order corrections to the superpotential). This construction allows one to geometrically construct mirrors of essentially arbitrary hypersurfaces in toric varieties. We will focus on examples such as pairs of pants and curves of arbitrary genus.