Various notions of multivariate negative dependence are introduced and their interrelationship is studied. Examples are given to illustrate these concepts. Applications of the results in statistics and probability are given. A partial ordering is developed among negative quadrant dependent distributions with fixed marginals. Basic properties and closure under certain statistical operations are derived. Applications of the results in statistics and probability are given.;Various definitions of multivariate new better than used (NBU) and new better than used in expectation (NBUE) life distributions are introduced and their interrelationship is studied. Examples are given to illustrate these concepts. Various closure properties of multivariate NBU and NBUE distributions are proved. Finally, it is shown how shock models governed by a general counting process satisfying certain conditions can generate multivariate NBU and NBUE distributions.