  This work gives a manual for constructing superconformal field theories
associated to a family of smooth K3 surfaces. A direct method is not known, but
a combination of orbifold techniques with a non-classical duality turns out to
yield such models. A four parameter family of superconformal field theories
associated to certain quartic K3 surfaces in CP^3 is obtained, four of whose
complex structure parameters give the parameters within superconformal field
theory. Standard orbifold techniques are used to construct these models, so on
the level of superconformal field theory they are already well understood.
  All "very attractive" K3 surfaces belong to the family of quartics underlying
these theories, that is all quartic hypersurfaces in CP^3 with maximal Picard
number whose defining polynomial is given by the sum of two polynomials in two
variables. A particular member of the family is the (2)^4 Gepner model, such
that these theories can be viewed as complex structure deformations of (2)^4 in
its geometric interpretation on the Fermat quartic.
