  We present a generalization of the complete intersection in products of
projective space (CICY) construction of Calabi-Yau manifolds. CICY three-folds
and four-folds have been studied extensively in the physics literature. Their
utility stems from the fact that they can be simply described in terms of a
`configuration matrix', a matrix of integers from which many of the details of
the geometries can be easily extracted. The generalization we present is to
allow negative integers in the configuration matrices which were previously
taken to have positive semi-definite entries. This broadening of the complete
intersection construction leads to a larger class of Calabi-Yau manifolds than
that considered in previous work, which nevertheless enjoys much of the same
degree of calculational control. These new Calabi-Yau manifolds are complete
intersections in (not necessarily Fano) ambient spaces with an effective
anticanonical class. We find examples with topology distinct from any that has
appeared in the literature to date. The new manifolds thus obtained have many
interesting features. For example, they can have smaller Hodge numbers than
ordinary CICYs and lead to many examples with elliptic and K3-fibration
structures relevant to F-theory and string dualities.
