In this paper, we study the problem of large deviations for measures with random weights. We are motivated by previous work dealing with the special case occuring in the statistical mechanics of the Bose gas. We study the problem in an abstract setting, isolating what is general from what is dependent on Bose statistics. We succeed in proving the large deviation principle for a large class of measures with random weights and obtaining the corresponding rate function in an explicit form. In particular, our results are applicable to the Fermi gas and the spherical model.