  Using several complex variables techniques, we investigate the interplay
between the geometry of the boundary and compactness of Hankel operators. Let f
be a function smooth up to the boundary on a smooth bounded pseudoconvex domain
D in C^n. We show that, if D is convex or the Levi form of the boundary of D is
of rank at least n-2, then compactness of the Hankel operator H_f implies that
f is holomorphic "along" analytic discs in the boundary. Furthermore, when D is
convex in C^2 we show that the condition on f is necessary and sufficient for
compactness of H_f
