We consider the problem of characterizing graphs with low ply number and algorithms for creating layouts of graphs with low ply number. Informally, the ply number of a straight-line drawing of a graph is defined as the maximum number of overlapping disks, where each disk is associated with a vertex and has a radius that is half the length of the longest edge incident to that vertex. We show that internally triangulated biconnected planar graphs that admit a drawing with ply number 1 can be recognized in O(n log n) time, while the problem is in general NP-hard. We also show several classes of graphs that have 1-ply drawings. We then show that binary trees, stars, and caterpillars have 2-ply drawings, while general trees have (h+1)-ply drawings, where h is the height of the tree. Finally we discuss some generalizations of the notion of a ply number.