We have studied the influence of weak lateral adsorbate-adsorbate interactions on submonolayer thermal desorption spectra using a comprehensively tested novel Kinetic Monte Carlo (KMC) simulation scheme. Both attractive and repulsive adsorbate-adsorbate interactions are investigated. It is shown, that the desorption order continuously changes from zero to first to second order like behavior when decreasing the adsorbate-adsorbate binding energy from positive (attractive) to negative (repulsive) values. The desorption of N-2 on Cu(1 1 0)-(2 x 1)O and Xe on Pt(1 1 1) provide examples for systems crucially influenced by weak lateral interactions. Both systems are investigated by means of KMC simulations, which allow to quantitatively reproduce the experimental desorption spectra. (C) 2003 Elsevier Science B.V. All rights reserved.