Abstract: The structure of the proton and other hadrons can be expressed in various ways, e.g. by considering non-local correlations between its constituents, or the distribution of fundamental properties like charge and energy inside it, which are encoded in its parton distribution functions (PDFs) and its form factors respectively. Fully characterizing hadron structure is important not only to properly describe how it emerges from the degrees of freedom of quantum chromodynamics (QCD), but also because it is input and sometimes a major source of uncertainty in collider searches for new physics. Lattice QCD is the only approach to theoretically predict these quantities from first principles. I will present recent lattice studies that either aim to increase precision of well-studied hadron properties like the nucleon axial charge and PDFs via improved techniques, or to provide the first glimpse into new ways to probe hadron structure, e.g. via the gravitational form factors.