  We describe the Szeg\"o kernel on a higher genus Riemann surface in terms of
Szeg\"o kernel data coming from lower genus surfaces via two explicit sewing
procedures where either two Riemann surfaces are sewn together or a handle is
sewn to a Riemann surface. We consider in detail the examples of the Szeg\"o
kernel on a genus two Riemann surface formed by either sewing together two
punctured tori or by sewing a twice-punctured torus to itself. We also consider
the modular properties of the Szeg\"o kernel in these cases.
