The authors survey some of the methods that have been used to study Euler sums, and they introduce a powerful new approach. They apply residue calculus to integrals of the form
where is the limit of integrals taken along large circles that expand to , is a rational function that is for large , and is a kernel function that is on large circles whose radii tend to . By employing kernels that are polynomials in , its derivatives and related trigonometric functions, they deduce a host of known relations on Euler sums and discover many new ones. A modification also gives results on alternating Euler sums.