In space plasmas, e.g., planetary magnetospheres and the solar wind, it has been observed that particle velocity distributions typically possess a non-Maxwellian high-energy tail that can be well modeled by a generalized Lorentzian (kappa) distribution. The generalized Lorentzian distribution is characterized by a spectral index &kgr;, varies as {energy}-(&kgr;+1) at high velocities, and approaches a Maxwellian distribution as &kgr;→∞. As a natural analogue to the widely used plasma dispersion function Z(&xgr;), which is based on the Maxwellian distribution, we have recently introduced a new special function Z&kgr;*(&xgr;) based on the generalized Lorentzian distribution; we call Z&kgr;*(&xgr;) the modified plasma dispersion function. Because Z&kgr;*(&xgr;) can be expressed in simple closed form, Z&kgr;*(&xgr;) is easier to use than Z(&xgr;) both from analytical and computational points of view. Z&kgr;*(&xgr;) is, moreover, a natural took for analyzing microinstabilities in a variety of space plasmas. In this paper we use Z&kgr;*(&xgr;) to analyze three classical problems of plasma physics: Landau damping of Langmuir waves; ion acoustic instability in a current-carrying plasma; and cyclotron resonant instability of electromagnetic R mode waves propagating parallel to an ambient magnetic field. In each case we find that results for a generalized Lorentzian plasma can differ significantly from those in a Maxwellian plasma. Previous calculations based on a Maxwellian distribution, that purport to apply to waves in space, may therefore be subject to reexamination. ¿ American Geophysical Union 1992