How are moments and moments of spacings related to distribution functions?

It is shown how rth moments of random variables and rth product moments of spacings between random variables can be written as r-dimensional integrals involving only the distribution function, and not the density or any other function of the variables of integration. These fundamental formulae are the main focus of the paper and appear to be new for r⩾3, as well as for the second cross-moment of spacings. The formulae for ordinary moments are derived separately for r=3 and 4. Those for moments of spacings allow an expression for quite general r. Links with L-moments and with expressions for the moments of the sample range are explored. Extensions to the independent but not identically distributed case are given.