Pulse-echo-overlap measurements of ultrasonic wave velocity have been used to determine the elastic stiffness moduli and related elastic properties of aluminum nitride (AlN) ceramic samples as functions of temperature in the range 100-295 K and hydrostatic pressure up to 0.2 GPa at room temperature. Aluminum nitride is an elastically stiff but light ceramic: at 295 K, the longitudinal stiffness (C-L), shear stiffness (mu), adiabatic bulk modulus (B-S), Young's modulus (E) and Poisson's ratio (sigma) are 373 GPa, 130 GPa, 200 GPa, 320 GPa and 0.234, respectively. The temperature dependences of C-L and B-S show normal behaviour and can be approximated by the conventional model for vibrational anharmonicity. The results of measurements of the effects of hydrostatic pressure on the ultrasonic wave velocity have been used to determine the hydrostatic-pressure derivatives of elastic stiffnesses and the acoustic-mode Gruneisen parameters. The values determined at 295 K for the hydrostatic-pressure derivatives (partial derivativeC(L)/partial derivativeP)(P=0), (partial derivative mu/partial derivativeP)(P=0) and (partial derivativeB(S)/partial derivativeP)(P=0) are 4.7 +/- 0.1, 0.22 +/- 0.03 and 4.4 +/- 0.15, respectively. The adiabatic bulk modulus B-S and its hydrostatic-pressure derivative (partial derivativeB(S)/partial derivativeP)(P=0) are in good agreement with the results of recent high pressure X-ray diffraction measurements and theoretical calculations. The longitudinal (gamma (L)), shear (gamma (S)), and mean (gamma (el)) acoustic-mode Gruneisen parameters of AlN are positive: the zone-centre acoustic phonons stiffen under pressure. The shear gamma (S) (=0.006) is much smaller than the longitudinal gamma (L) (=1.09) accounting for the low thermal Grneisen parameter gamma (th) (=0.65) obtained for this ceramic: since the acoustic Debye temperature Theta (D) (=980 +/- 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 AlN.