Abstract: Let K be a finite simplicial complex with an explicit piecewise Euclidean metric assigned to each simplex so that these metrics agree under restriction. One of the first questions a geometric group theorist would ask in a situation like this is whether the complex K is in fact nonpositively curved. Even when K is finite and the metric is given, there are very few known cases in which this question can be resolved one way or the other in an algorithmic fashion. In this talk I will review the cases that are known and I will describe some recent progress on this question. (Joint work with Murray Elder).