  We discuss the Green-Schwarz action for type IIB strings in general
plane-wave backgrounds obtained as Penrose limits from any IIB supergravity
solutions with vanishing background fermions. Using the normal-coordinate
expansion in superspace, we prove that the light-cone action is necessarily
quadratic in the fermionic coordinates. This proof is valid for more general
pp-wave backgrounds under certain conditions. We also write down the complete
quadratic action for general bosonic on-shell backgrounds in a form in which
its geometrical meaning is manifest both in the Einstein and string frames.
When the dilaton and 1-form field strength are vanishing, and the other field
strengths are constant, our string-frame action reduces, up to conventions, to
the one which has been written down using the supercovariant derivative.
