Summary: A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit length line segments is called a unit-distance graph. In general, the number of unit edges in a graph depends on the placement of its vertices in the plane. In this paper we consider graphs having the maximum number of unit edges. Three cases are considered: (i) all non-unit edges shorter than unit edges; (ii) all non-unit edges longer than unit edges; and (iii) non-unit edges either shorter or longer than unit edges. Some results are obtained for $K_{m,n}$, and several open questions are discussed.