Antibodies are widely used in biology and medicine, and there has been considerable interest in multivalent antibody formats to increase binding avidity and enhance signaling pathway agonism. However, there are currently no general approaches for forming precisely oriented antibody assemblies with controlled valency. We describe the computational design of two-component nanocages that overcome this limitation by uniting form and function. One structural component is any antibody or Fc fusion and the second is a designed Fc-binding homo-oligomer that drives nanocage assembly. Structures of 8 antibody nanocages determined by electron microscopy spanning dihedral, tetrahedral, octahedral, and icosahedral architectures with 2, 6, 12, and 30 antibodies per nanocage match the corresponding computational models. Antibody nanocages targeting cell-surface receptors enhance signaling compared to free antibodies or Fc-fusions in DR5-mediated apoptosis, Tie2-mediated angiogenesis, CD40 activation, and T cell proliferation; nanocage assembly also increases SARS-CoV-2 pseudovirus neutralization by α-SARS-CoV-2 monoclonal antibodies and Fc-ACE2 fusion proteins. We anticipate that the ability to assemble arbitrary antibodies without need for covalent modification into highly ordered assemblies with different geometries and valencies will have broad impact in biology and medicine.