Graphs, in the sense of vertices and edges, are ubiquitous mathematical models for systems involving binary relationships. They may also represent systems that have physical reality. To visualize a graph, or to realize it as a circuit of wires or a network of pipes or highways, we must assign its vertices and edges to concrete locations: we must "draw" the graph.

The increasing complexity of our world presents new challenges for graph layout, as this talk will demonstrate. We will review the historical roots of graph layout and embedding problems, move on to computational aspects of these problems and how they may be dealt with, and describe some current challenges along the way.