A graph is chordal if there are no induced cycles of length 4 or more. A chordal completion of a graph is formed by adding edges until the resulting graph is chordal. What is the minimal number of edges in a chordal completion? The authors answer this question for the class of planar graphs: every planar graph on \(n\) vertices has a chordal completion with \(cn\log n\) edges for some absolute constant \(c\). Moreover, the result is best possible (up to the constant) as shown by the \(n\) by \(n\) grid. The authors also prove that any chordal completion of the \(n\) by \(n\) grid must contain a clique of size \(cn\) for some absolute constant \(c\). The proof of the main result follows by examining the size of the smallest set of vertices which disconnects a planar set into two parts of roughly equal size. That the result is best possible follows from a counting argument on cycles in the grid. The result about cliques in chordal completion of grids follows from a relationship with tree- width. The authors give a few results about chordal completions for other types of graphs, most notably the family of graphs of bounded genus and random graphs. An interesting concluding section focuses on applications to artificial intelligence, most notably computer vision.