  We establish an explicit characterisation of L\'evy measures on both
$L^p$-spaces and UMD Banach spaces. In the case of $L^p$-spaces, L\'evy
measures are characterised by an integrability condition, which directly
generalises the known description of L\'evy measures on sequence spaces. The
latter has been the only known description of L\'evy measures on infinite
dimensional Banach spaces that are not Hilbert. L\'evy measures on UMD Banach
spaces are characterised by the finiteness of the expectation of a random
{\gamma}-radonifying norm. Although this description is more abstract, it
reduces to simple integrability conditions in the case of $L^p$-spaces.
