We determine the Euclidean distance degree of a projective toric variety. This extends the formula of Matsui and Takeuchi for the degree of the A-discriminant in terms of Euler obstructions. The motivation for this work is the development of reliable methods for computing the points on a real toric variety that are closest to a given data point. In this lecture we emphasise the role played by characteristic classes such as the Chern-Mather class. This is joint work with Martin Helmer.