Abstract: In modern risk management, a crucial consideration is how to describe and quantify a risk according to its tail behaviour. Prominent, yet elementary, examples of tail risk measures are Value-at-Risk (VaR) and the Expected Shortfall (ES), which are two popular classes of regulatory risk measures in banking and insurance. To achieve a comprehensive understanding of the tail risk, we carry out an axiomatic study for risk measures which quantify the tail risk, that is, the behavior of a risk beyond a certain quantile. We establish a connecting between a tai risk measure and its generator, and investigate their joint properties. A tail risk measure inherits many properties from its generator, but not subadditivity or convexity; nevertheless, a tail risk measure is coherent if and only if its generator is coherent. We explore further tail shortfall risk measure and elicitability. In particular, there is no elicitable tail convex risk measure rather than the essential supremum, and under a continuity condition, the only elicitable and positively homogeneous monetary tail risk measures are the VaRs.