  The aim of this paper is to construct and analyze explicit exponential
Runge-Kutta methods for the temporal discretization of linear and semilinear
integro-differential equations. By expanding the errors of the numerical method
in terms of the solution, we derive order conditions that form the basis of our
error bounds for integro-differential equations. The order conditions are
further used for constructing numerical methods. The convergence analysis is
performed in a Hilbert space setting, where the smoothing effect of the
resolvent family is heavily used. For the linear case, we derive the order
conditions for general order $p$ and prove convergence of order $p$, whenever
these conditions are satisfied. In the semilinear case, we consider in addition
spatial discretization by a spectral Galerkin method, and we require locally
Lipschitz continuous nonlinearities. We derive the order conditions for orders
one and two, construct methods satisfying these conditions and prove their
convergence. Finally, some numerical experiments illustrating our theoretical
results are given.
