Understanding interspecific viral transmission is key to understanding viral ecology and evolution, disease spillover into humans, and the consequences of global change. Prior studies have uncovered macroecological drivers of viral sharing, but analyses have never attempted to predict viral sharing in a pan-mammalian context. Using a conservative modelling framework, we confirm that host phylogenetic similarity and geographic range overlap are strong, nonlinear predictors of viral sharing among species across the entire mammal class. Using these traits, we predict global viral sharing patterns of 4196 mammal species and show that our simulated network successfully predicts viral sharing and reservoir host status using internal validation and an external dataset. We predict high rates of mammalian viral sharing in the tropics, particularly among rodents and bats, and within- and between-order sharing differed geographically and taxonomically. Our results emphasize the importance of ecological and phylogenetic factors in shaping mammalian viral communities, and provide a robust, general model to predict viral host range and guide pathogen surveillance and conservation efforts.