Measurements of the temperature dependences of the hydrostatic-pressure derivatives of the velocities of ultrasonic waves propagated in single crystals of the Invars Fe 72 Pt 28 and Fe 72 Pt 25 Ni 3 verify that the negative thermal expansion in the ferromagnetic phase of these alloys is directly associated with longitudinal-acoustic-mode softening. In the paramagnetic phase of Fe 72 Pt 28 , the hydrostatic-pressure derivatives of each of the elastic-tensor components and the bulk modulus B are positive, showing normal behavior in the sense that the long-wavelength acoustic-phonon frequencies increase under pressure. However, below the Curie temperature T C the velocities of longitudinal untrasonic waves propagated along the [100] and [110] directions in Fe 72 Pt 28 and Fe 72 Pt 25 Ni 3 decrease strongly with pressure; thus (∂ C 11 /∂P ) P = 0 , (∂ C L /∂P ) P = 0 , and (∂ B S /∂P ) P = 0 are negative due to the magnetoelastic interaction. These Invar alloys show the extraordinary property of becoming easier to compress as the pressure is increased. The negative signs of (∂ C 11 /∂P ) P = 0 and (∂ C L /∂P ) P = 0 give rise to negative values for all the longitudinal- and quasilongitudinal-acoustic-mode Grüneisen parameters in the ferromagnetic phase. This experimental observation is in accord with a recent prediction of negative longitudinal-acoustic-mode Grüneisen parameters stemming from itinerant-electron-magnetism theory. For Fe 72 Pt 28 the hydrostatic-pressure derivative (∂ C 11 /∂P ) P = 0 is negative, attains its maximum value just above room temperature, and becomes much smaller as the temperature is lowered, matching, and accounting for, the behavior of the thermal expansion, which is negative in the temperature range between about 260 K and the Curie temperature. In the case of the archetypal Invar alloy Fe 65 Ni 35 , the hydrostatic-pressure derivatives of the elastic-stiffness-tensor components are positive, but (∂ C 11 /∂P ) P = 0 and (∂ C L /∂P ) P = 0 are small in the ferromagnetic phase, consistent with its small but positive thermal expansion. It is concluded that longitudinal-acoustic-mode softening due to the magnetoelastic interaction is the source of the Invar behavior of each of these iron alloys. In addition, measurements of the temperature dependences of the ultrasonic wave velocities establish that the fcc-bct martensitic phase transition in Fe-Ni and Fe-Pt alloys is driven by a soft shear zone-center acoustic phonon with propagation vector 〈110〉 and polarization vector 〈11¯0〉.