  Based on a recent work on traveling waves in spatially nonlocal
reaction-diffusion equations, we investigate the existence of traveling fronts
in reaction-diffusion equations with a memory term. We will explain how such
memory terms can arise from reduction of reaction-diffusion systems if the
diffusion constants of the other species can be neglected. In particular, we
show that two-scale homogenization of spatially periodic systems can induce
spatially homogeneous systems with temporal memory.
  The existence of fronts is proved using comparison principles as well as a
reformulation trick involving an auxiliary speed that allows us to transform
memory terms into spatially nonlocal terms. Deriving explicit bounds and
monotonicity properties of the wave speed of the arising travelingfront, we are
able to establish the existence of true traveling fronts for the original
problem with memory. Our results are supplemented by numerical simulations.
