For $n\in \mathbb{N}$ and $D\subseteq \mathbb{N}$, the distance graph $P_n^D$ has vertex set $\{ 0,1,\ldots,n-1\}$ and edge set $\{ ij\mid 0\leq i,j\leq n-1, |j-i|\in D\}$. The class of distance graphs generalizes the important and very well-studied class of circulant graphs which have been proposed for numerous network applications. In view of fault tolerance and delay issues in these applications, the connectivity and diameter of circulant graphs have been studied in great detail. Our main contributions are hardness results concerning computational problems related to the connectivity and diameter of distance graphs and a number-theoretic characterization of the connected distance graphs $P_n^D$ for $|D|=2$.