Toeplitz posed the following question in 1911: On every Jordan curve in R^2, are there four points which form a perfect square? In the intervening years, a string of authors have published proofs of various cases of the problem, leading to the results of W. Stromquist and H.B. Griffiths in the 1990's that inscribed squares exist on certain classes of Jordan curves which are somewhat more general than C^1 curves. We present a new approach to the problem which provides a collection of new results and new proofs of existing theorems, including a new proof of the square peg theorem itself. Here is another example of our results: Given a point p on a closed C^1 curve C in a Riemannian manifold M and any n, there is an equilateral geodesic n-gon inscribed in C with a vertex at p. The key idea in the new approach is to rethink the problem in terms of intersection numbers and Dev Sinha's geometric reformulation of the compactified configuration spaces of Fulton-MacPherson. This talk represents joint work with Elizabeth Denne (Harvard) and John McCleary (Vassar College).