Abstarct: In this thesis, the concept of tail subadditivity (Belles-Sampera et al., 2014a and 2014b) for distortion risk measures is extended. Also sufficient and necessary conditions for a distortion risk measure to be tail subadditive are given. The generalized GlueVaR risk measures, which can be used to approach any coherent distortion risk measure are introduced. To further illustrate the applications of the tail subadditivity, the multivariate tail distortion (MTD) risk measures are proposed and the multivariate tail conditional expectation (MTCE) risk measure introduced by Landsman et al. (2016) is generalized. The properties of multivariate tail distortion risk measures, such as positive homogeneity, translation invariance, monotonicity, and subadditivity, are discussed as well. Moreover, the applications of the multivariate tail distortion risk measures in capital allocations for a portfolio of risks are discussed and the impacts of the dependence between risks in a portfolio and extreme tail events of a risk portfolio in capital allocations are explored.