We study enumerative aspects of moduli space of sheaves on smooth quasi-projective varieties of dimension three and four. Our focus is on moduli spaces of rank one sheaves and complexes, including Donaldson-Thomas theory and Pandharipande-Thomas’s stable pairs. Unlike the moduli space of sheaves on curves, the moduli space of sheaves on higher dimensional varieties can have arbitrary singularities, making the study of its geometry difficult. Nevertheless, there remains a method for constructing virtual cycles that enables to study of enumerative aspects.