How strongly can electromagnetic radiation interact with a linear, passive scattering body? Here I show that passivity—the condition the polarization currents do no work—in conjunction with the optical theorem imposes convex constraints on the currents (real or effective) that can be excited in any scatterer, leading to general shape- and/or material-independent bounds. One ramification is the prospect of hybrid dielectric–metal plasmonic structures that outperform even the theoretical limits for all-metal approaches. A key degree of freedom in the passivity constraints is the arbitrary choice of “incident” and “scattered” fields, which enables the first bounds to radiative heat transfer in the near field.