Understanding the gluonic composition of hadrons and nuclei is the next frontier for hadron structure and a 
major motivator for the EIC. I will discuss lattice QCD calculations of moments of gluonic (generalised)
parton distributions in hadrons and nuclei that aim to provide Standard Model benchmarks for quantities that
can be measured for the first time at the EIC. I will focus on the gluonic transversity distribution that provides a 
clean probe of gluonic structure as it does not mix with quark operators until higher-twist, and on possible 
signatures for a gluonic analogue of the EMC effect in light nuclei.