Multi-component aggregates are being intensively researched in various fields because of their highly tunable properties and wide applications. Due to the complex configurational space of these systems, research would greatly benefit from a general theoretical framework for the prediction of stable structures, which, however, is largely incomplete at present. Here we propose a general theory for the construction of multi-component icosahedral structures by assembling concentric shells of different chiral and achiral types, consisting of particles of different sizes. By mapping shell sequences into paths in the hexagonal lattice, we establish simple and general rules for designing a wide variety of magic icosahedral structures, and we evaluate the optimal size-mismatch between particles in the different shells. The predictions of our design strategy are confirmed by molecular dynamics simulations and density functional theory calculations for several multi-component atomic clusters and nanoparticles.