  We give a complete description of the boundary behaviour of the Poisson
kernel and the harmonic Bergman kernel of a bounded domain with smooth
boundary, which in some sense is an analogue of the similar description for the
usual Bergman kernel on a strictly pseudoconvex domain due to Fefferman. Our
main tool is the Boutet de Monvel calculus of pseudodifferential boundary
operators, and in fact we describe the boundary singularity of a general
potential, trace or singular Green operator from that calculus.
