  We derive a component-field expansion of the Green-Schwarz action for the
type IIA string, in an arbitrary background of massless NS-NS and R-R bosonic
fields, up to quadratic order in the fermionic coordinates \theta. Using this
action, we extend the usual derivation of Buscher T-duality rules to include
not only NS-NS, but also R-R fields. Our implementation of the T-duality
transformation rules makes use of adapted background-field parametrizations,
which provide a more geometrically natural and elegant description for the
duality maps than the ones previously presented. These T-duality rules allow us
to derive the Green-Schwarz action for the type IIB string in an arbitrary
background of massless NS-NS and R-R bosonic fields, up to O(\theta^2).
Implemention of another T-duality transformation on this type IIB action then
allows us also to derive the Green-Schwarz action for the massive IIA string.
By further considering T-duality transformations for backgrounds with the two
U(1) isometries of a 2-torus, we give a string-theoretic derivation of the
direct T-duality relation between the massless and massive type IIA strings. In
addition, we give an explicit construction of the D=8 SL(3,R)xSL(2,R) invariant
supergravity with two mass parameters that form a doublet under the SL(2,R)
factor.
