The paper gives fundamental results on the universal abelian covers of rational surface singularities. Let $(X,o)$ be a normal complex surface singularity germ with a rational homology sphere link. Then $(X,o)$ has the universal abelian cover $(Y,o) \longrightarrow (X,o)$. It is shown that if $(X,o)$ is rational or minimally elliptic, and if it has a star-shaped resolution graph, then $(Y,o)$ is a complete intersection (a partial answer to the conjecture of Neumann and Wahl). A way is given to compute the multiplicity and the embedding dimension of $(Y,o)$ from the resolution graph of $(X,o)$ in the case when $(X,o)$ is rational.