Gravitational form factors (GFFs) parametrize the hadronic matrix elements of the energy-momentum tensor (EMT) of QCD and encode important information about hadrons, e.g., how energy and angular momentum are distributed inside them. They can be decomposed into quark and gluon parts, allowing disentanglement of the contribution of individual QCD degrees of freedom in hadron structure. GFFs have received a lot of interest in the hadron physics community since experimental constraints from JLab measurements have become available over the past 5 years. Lattice QCD can directly probe these quantities, offering complimentary information to current and future experimental programs, e.g., at the upcoming US Electron Ion Collider. In this talk, I will discuss the current status of GFF research, focusing on both the successes and the challenges of lattice QCD in constraining them. I will present recent results on the first lattice QCD computation of the flavor decomposition of the pion and the proton GFFs at near-physical pion mass and discuss their implications for hadron structure. Finally, I will present ongoing work on constraining the GFFs of glueballs in Yang-Mills theory.