The elastic and nonlinear acoustic vibrational properties of terbium metaphosphate glasses (Tb203)x(P205)1-x with x = 0·226, 0·247, 0·263 and 0·271 (x is the mole fraction) have been determined from measurements of the effects of temperature, hydrostatic pressure, and uniaxial stress on' ultrasonic wave velocity. At temperatures below about 140 K, the elastic stiffness C1J of' (Tb203)x(P205)1-x glasses becomes anomalously dependent upon temperature, a behaviour usually associated with interactions between acoustic phonons and two-level systems. Except for the (Tb203)O·271(P20S)O·729 glass, the hydrostatic pressure derivatives (8C1J/8P)T,P=O of the elastic stiffness C1J and also (8B s /8P)T,P=O of the bulk modulus B S of terbium metaphosphate glasses are small and negative. The third-order elastic stiffness tensor components CIJK of the (Tb203)o·247(P20s)o·7s3 glass between 77 K and 400 K have also been determined. At room temperature, Cl12, C123 and C144 are positive while C111, C155 and C456 are negative. Both longitudinal and shear acoustic mode Griineisen parameters are small and negative: the application of pressure softens the long-wavelength acoustic phonon mode frequencies. The mode softening is enhanced as the temperature is reduced.