The structures, phase transition, mechanical properties, dynamic stability, and electronic properties of SnO polymorphs (α, γ, herzenbergite, B1, and B2 phases) under pressure have been studied using the first‐principles calculations. The obtained structural parameters are in agreement with the available data. According to the enthalpy–pressure curves of SnO, the pressure‐induced phase transitions are verified to be from α to γ at 0.94 GPa and from B1 to B2 at 49.1 GPa, respectively. The calculated elastic constants indicate that B2 and γ are unstable under zero pressure. With increasing pressure, there is stability–instability transition for B1, whose transition pressure is 20.6 GPa, respectively. Moreover, a range of mechanical stability is from 5.5 to 10.8 for α phase and from 3.4 to 14.3 GPa for γ phase. The corresponding elastic modulus has been analyzed under pressures. With regard to elastic anisotropy, a variety of methods are taken to analyze it and the causes of various anisotropic characters are explained. Phonon dispersions show the same dynamic stability of SnO with the mechanical stability. In addition, the density of states and charge density all reflect that the interactions between Sn and O elements are enhanced with the increasing pressure.