The adsorption of buckminsterfullerene (C-60) on metal surfaces has been investigated extensively for its unique geometric and electronic properties. The two-dimensional systems formed on surfaces allow studying in detail the interplay between bonding and electronic structures. Recent studies reveal that C-60 adsorption induces reconstruction of even the less-reactive close-packed metal surfaces. First-principles computations enable access to this important issue by providing not only detailed atomic structure but also electronic properties of the substrate-adsorbate interaction, which can be compared with various experimental techniques to determine and understand the interface structures. This review discusses in detail the ordered phases of C-60 monolayers on metal surfaces and the surface reconstruction induced by C-60 adsorption, with an emphasis on the different types of reconstruction resulting on close-packed metal surfaces. We show that the symmetry matching between C-60 molecules and metal surfaces determines the local adsorption configurations, while the size matching between C-60 molecules and the metal surface lattice determines the supercell sizes and shapes; importantly and uniquely for C-60, the number of surface metal atoms within one supercell determines the different types of reconstruction that can occur. The atomic structure at the molecule-metal interface is of crucial importance for the monolayer's electronic and transport properties: these will also be discussed for the well-defined adsorption structures, especially from the perspective of tuning the electronic structure via C-60-metal interface reconstruction and via relative inter-C-60 orientations.