We have successfully grown InGaAs/AIAsSb quantum-well (QW) structures lattice-matched to InP by molecular beam epitaxy for the first time. We studied the band-edge discontinuity and the interface abruptness of these heterostructures. A cross-sectional lattice image of InGaAs/AlAsSb QWs taken along the  axis showed atomically smooth heterointerfaces. The photoluminescence (PL) peak energy of the 20-nm-thick InGaAs well (0.758 eV) was lower than that of InGaAs bulk (0.799 eV), indicating that the InGaAs/AlAsSb system has a staggered lineup. The conduction band-edge discontinuity, ΔEc, was evaluated to be 1.74 ± 0.04 eV, which was derived from parameter fitting to the 4.2 K PL peak energy shifts of QWs as a function of InGaAs well width between 2.1 nm and 20 nm. The corresponding valence band-edge discontinuity, ΔEv, was 0.07 ± 0.02 eV. We also fabricated a resonant tunneling barrier structure of InGaAs (4.4 nm)/AlAsSb (2.9 nm), and obtained a very high peak-to-valley current ratio of 15 at 300 K.