Exponential functionals of the form ò₀^t e^-x_s- dh_s \int_{0}^t \mathrm{e}^{-\xi_{s-}} \mathrm{d}\eta_s constructed from a two dimensional Lévy process (x,h)(\xi,\eta) are of interest and application in many areas. In particular, the question of the convergence of the integral ò₀^¥ e^-x_t- dh_t \int_{0}^\infty \mathrm{e}^{-\xi_{t-}} \mathrm{d}\eta_t arises in recent investigations such as those of Barndorff-Nielsen and Shephard [3] in financial econometrics, and in those of Carmona, Petit and Yor [9], and Yor [40, 41], where it is related among other things to the existence of an invariant measure for a generalised Ornstein-Uhlenbeck process. We give a complete solution to the convergence question for integrals of the form ò₀^¥ g(x_t-) dh_t\int_0^\infty g(\xi_{t-}) \mathrm{d}\eta_t , when g(t) = e^-t and h_t\eta_t is general, or g(·)g(\cdot) is a nonincreasing function and dh_t = d t\mathrm{d}\eta_t = \mathrm{d} t , and some other related results. The necessary and sufficient conditions for convergence are stated in terms of the canonical characteristics of the Lévy process. Some applications in various areas (compound Poisson processes, subordinated perpetuities, the Doléans-Dade exponential) are also outlined.