  The pure spinor formalism for the superstring can be formulated as a twisted
N=2 worldsheet theory with fermionic generators $j_{BRST}$ and composite $b$
ghost. After untwisting the formalism to an N=1 worldsheet theory with
fermionic stress tensor $j_{BRST}+b$, the worldsheet variables combine into N=1
worldsheet superfields $X^m$ and $\Theta^\alpha$ together with a superfield
constraint relating $DX^m$ and $D\Theta^\alpha$. The constraint implies that
the worldsheet superpartner of $\theta^\alpha$ is a bosonic twistor variable,
and different solutions of the constraint give rise to the pure spinor or
extended RNS formalisms, as well as a new twistor-string formalism with
manifest N=1 worldsheet supersymmetry.
  These N=1 worldsheet methods generalize in curved Ramond-Ramond backgrounds,
and a manifestly N=1 worldsheet supersymmetric action is proposed for the
superstring in an $AdS_5\times S^5$ background in terms of the twistor
superfields. This $AdS_5\times S^5$ worldsheet action is a remarkably simple
fermionic coset model with manifest $PSU(2,2|4)$ symmetry and might be useful
for computing $AdS_5\times S^5$ superstring scattering amplitudes.
