A Molecular Dynamics simulation study has been carried out of the Lennard–Jones fluid in the stable liquid and metastable liquid–vapour co‐existence regions. The interest has been mainly in states for which the pressure of the fluid is negative, and how they compare with those at positive pressure. The behaviour of the LJ fluid shows parallels with that of water, with the maximum tensile strength near the triple point being approximately seven times the critical pressure. The maximum tensile strength increases with cooling. Many of the properties, such as elastic moduli, show a smooth transition across the liquid–vapour boundary, on the liquid side, and a monotonic variation in the metastable region of the phase diagram. There is however a similarity in behaviour of combinations of the mechanical properties and the equation of state; for example, the pressure and infinite frequency Poissons ratio have minima at approximately the same densities. Insights into the heterogeneity of the assembly of particles are provided by probability distributions of the net force on a molecule and that of a definition for the local pressure experienced by a molecule, or ‘pressurelet'. The maximum negative pressure coincides with the maximum fraction of molecules individually experiencing a negative pressure. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)