  In this paper, we introduce new enumerative invariants of curves on
Calabi-Yau 3-folds via certain stable objects in the derived category of
coherent sheaves. We introduce the notion of limit stability on the category of
perverse coherent sheaves, a subcategory in the derived category, and construct
the moduli spaces of limit stable objects. We then define the counting
invariants of limit stable objects using Behrend's constructible functions on
that moduli spaces. It will turn out that our invariants are generalizations of
counting invariants of stable pairs introduced by Pandharipande and Thomas. We
will also investigate the wall-crossing phenomena of our invariants under
change of stability conditions.
