I will review an approach to the covariant quantization of superstrings in
Calabi-Yau backgrounds known as the hybrid formalism which arises as a
field redefinition of the standard RNS formalism. I will describe how
these models fit into the complete picture of fully covariant quantum
superstrings and point out a gap in the list. I will then fill the gap by
presenting a new hybrid formalism which describes the heterotic string on
the product of a K3 surface and a 2-torus.
