We describe the kernel and the image of the continuous extension of the boundary principal symbol to the norm closure of the algebra of all operators of order and class zero in Boutet de Monvel's algebra on a compact manifold with boundary. If the manifold is connected and the boundary is not empty, we then show that the K-groups of the quotient of that algebra by the compact ideal sit in the middle of short exact sequences of abelian groups determined by the topology of the manifold and its tangent bundle. If the K-groups of the manifold and of the tangent bundle of its interior are free and finitely generated (with Ki-dimensions equal to ki and li), then so are those of that quotient (with dimension k0+l1 for K0, and k1+l0 for K1). This talk is based on joint work with R. Nest and E. Schrohe.