We consider semilinear integrodifferential equations of the form \[ u'(t) + A(t)u(t) = \int_0^t {\left[ {a(t,s)g_0 (s,u(s)) + g_1 (t,s,u(s))} \right]ds + f_0 (t) + f_1 (t,u(t)),} \]\[ u(0) = u_0 . \] For each $t \geqq 0$, the operator $A(t)$ is assumed to be the negative generator of an analytic semigroup in a Banach space X. Thus, our models are Volterra integrodifferential equations of parabolic type. These types of equations arise naturally in the study of heat flow in materials with memory. Our main results are the proofs of local and global existence, uniqueness, continuous dependence and differentiability of solutions.