We show how special forms of an N=2 Landau-Ginzburg potential directly
imply the presence of an N=2 super-W algebra. If the Landau-Ginzburg
model has a super-W algebra, we show how the elliptic genus can be ref
ined so as to give much more complete information about the structure
of the model. We study the super-W3 model in some detail, and present
some results and conjectures about more general models.