The class of law invariant coherent risk measures contains many risk measures that one would encounter in a distribution setting. In this talk, we present some general results about worst-case law invariant risk measures where a set of distributions sharing the same first few moments are considered for estimating the worst possible risk. In particular, its distributionally robust optimization (DRO) formulation is generally nonlinear in distribution and thus requires additional care in studying its tractability. We show cases where worst-case risk measures and distributions admit closed-form expressions and discuss their implication for future research. Our analysis exploits the structure of spectral risk measure and its connection to law invariant risk measures.