  This paper initiates the study of fractional eternal domination in graphs, a
natural relaxation of the well-studied eternal domination problem. We study the
connections to flows and linear programming in order to obtain results on the
complexity of determining the fractional eternal domination number of a graph
$G$, which we denote $\gamma_{\,\textit{f}}^{\infty}(G)$. We study the
behaviour of $\gamma_{\,\textit{f}}^{\infty}(G)$ as it relates to other
domination parameters. We also determine bounds on, and in some cases exact
values for, $\gamma_{\,\textit{f}}^{\infty}(G)$ when $G$ is a member of one of
a variety of important graph classes, including trees, split graphs, strongly
chordal graphs, Kneser graphs, abelian Cayley graphs, and graph products.
