Summary: Let \(G\) be a simple, connected graph on \(n\) vertices. Let \(d_G(u,v)\) denote the distance between vertices \(u\) and \(v\) in \(G\). A subgraph \(H\) of \(G\) is isometric if \(d_H(u,v)=d_G(u,v)\) for every \(u\), \(v\) in V(H). We say that \(G\) is a distance-preserving graph if \(G\) contains at least one isometric subgraph of order \(k\) for every \(k\), \(1\leq k\leq n\). In this paper we construct regular distance-preserving graphs of all possible orders and degrees of regularity. By modifying the Havel-Hakimi algorithm, we are able to construct distance preserving graphs for certain other degree sequences as well. We include a discussion of some related conjectures which we have computationally verified for small values of \(n\).