In biology and mathematics compartmental systems are frequently used. System identification of systems based on physical laws often involves parameter estimation. Before parameter estimation can take place, we have to examine whether the parameters are structurally identifiable. In this paper tests for the structural identifiability of linear compartmental systems are proposed. The method is based on the similarity transformation approach. New contributions in the theory are the conditions for structural identifiability of structured positive linear systems. In addition, structural identifiability from the Markov parameters is extended to structural identifiability from the input-output data, in which the initial condition is (partially) unknown and nonnegligible. Finally, conditions are presented for structural identifiability of a sampled continuous-time linear dynamic system.