Abstract: Let M be a smooth compact oriented manifold in C^{n}. When does there exist a complex analytic variety having M as its boundary? In 1975 R. Harvey and B. Lawson solved this problem using a condition on the tangent spaces to M. In recent years H. Alexander and I gave a topological condition on M which can be viewed as a generalization of the argument principle of classical function theory. I shall talk about our solution.