This paper discusses the notion of tail risk, and the ability of a tail risk measure to reflect this kind of risk. In particular, Yamai and Yoshiba's (2001, 2002) notion of strict risk measure tail risk is discussed and linked with a different notion of tail risk, the pK-tail risk, which is the risk associated with the probability measure conditional on the event that the losses are at least as large as K. A subset of pK-tail risk measures that are free of strict risk measure tail risk is introduced. These notions are then extended to Yaari's (1987) dual theory and the distorted risk measures framework.