Ultrasonic wave velocity measurements have been used to determine the elastic stiffness moduli and related elastic properties of high-purity, dense beta -Si3N4 ceramic samples as functions of temperature in the range 150-295 K and hydrostatic pressure up to 0.2 GPa at room temperature. Due to its covalently bonded, rigid structural framework beta -Si3N4 is an elastically stiff material; the elastic stiffness moduli of the ceramic at 295 K are: C-L = 396 GPa, mu = 119 GPa, B-S = 238 GPa, E = 306 GPa, Poisson's ratio sigma = 0.285. The longitudinal elastic stiffness C-L increases with decreasing temperature and shows a knee at about 235 K; the decrease in slope below the knee indicates mode softening. The shear elastic stiffness mu shows mode softening which results in a plateau centred at about 235 K and an anomalous decrease with further reduction in temperature. The hydrostatic-pressure derivatives of elastic stiffnesses at 295 K are (partial derivativeC(L)/partial derivativeP)(P)=0 = 4.5 +/- 0.1, (partial derivativeB(S)/partial derivativeP)(P)=0 = 4.3 +/- 0.1 and (partial derivative mu/partial derivativeP)(P)=0 = 0.17 +/-0.02 (pressure < 0.12 GPa). An interesting feature of the nonlinear acoustic behaviour of this ceramic is that, in the pressure range above 0.12 GPa, the values obtained for (partial derivative mu/partial derivativeP)(P)=0 and the shear mode Gruneisen parameter (gamma (S)) are small and negative, indicating acoustic-mode softening under these higher pressures. Both the anomalous temperature and pressure dependences of the shear elastic stiffness indicate incipient lattice shear instability. The shear gamma (S)(=0.005) is much smaller than the longitudinal gamma (L)(=1.18) accounting for the thermal Gruneisen parameter gamma (t)h(=1.09): since the acoustic Debye temperature Theta (D)(=923 +/- 5 K) is so high, the shear modes play an important role in acoustic phonon population at room temperature. Hence knowledge of the elastic and nonlinear acoustic properties sheds light on the thermal properties of ceramic beta -Si3N4.