2D materials provide a platform for strong light-matter interactions, creating wide-ranging design opportunities via new-material discoveries and new methods for geometrical structuring. We derive general upper bounds to the strength of such light-matter interactions, given only the optical conductivity of the material, including spatial nonlocality, and otherwise independent of shape and configuration. Our material figure of merit shows that highly doped graphene is an optimal material at infrared frequencies, whereas single-atomic-layer silver is optimal in the visible. For quantities ranging from absorption and scattering to near-field spontaneous-emission enhancements and radiative heat transfer, we consider canonical geometrical structures and show that in certain cases the bounds can be approached, while in others there may be significant opportunity for design improvement. The bounds can encourage systematic improvements in the design of ultrathin broadband absorbers, 2D antennas, and near-field energy harvesters.2D materials [1, 2] and emerging methods [3][4][5][6][7][8] for patterning 2D layers and their surroundings are opening an expansive design space, exhibiting significantly different optical [9][10][11] (and electronic) properties from their 3D counterparts. In this Letter, we identify energy constraints embedded within Maxwell's equations that impose theoretical bounds on the largest optical response that can be generated in any 2D material, in the near or far field. The bounds account for material loss as encoded in the real part of a material's conductivityin the case of a spatially local conductivity tensor σ, they are proportional to σ † (Re σ) −1 σ -and are otherwise independent of shape and configuration. We derive the bounds through convex constraints imposed by the optical theorem [12][13][14] and its near-field analogue, leveraging a recent approach we developed for spatially local 3D materials [15]. In addition to accommodating nonlocal models, this work demonstrates starkly different near-field dependencies of 2D and 3D materials. For graphene, the 2D material of foremost interest to date, the bounds bifurcate into distinctive low-and highenergy regimes: the low-energy bounds are proportional to the Fermi level, whereas the high-energy bounds are proportional to the fine-structure constant, α, for any geometrical configuration. We find that far-field bounds on the extinction cross-section can be approached by elliptical graphene disks, whereas the near-field bounds on the local density of states [16][17][18][19][20] and radiative heat transfer rate [21][22][23][24][25][26] cannot be approached in prototypical flatsheet configurations. The bounds presented here provide a simple material figure of merit to evaluate the emerging zoo of 2D materials, and offer the prospect of greater optical response via computational design. The material * Corresponding author: owen.miller@yale.edu figure of merit can guide ongoing efforts in 2D-material discovery, while the general bounds can shape and ...