Using the first-principles plan-wave pseudo-potential method, a systematic investigation on structural property, magnetism, and pressure-induced phase transitions of cobalt is carried out. Cobalt with our hcp (α), fcc (β), bcc (γ), and epsilon (ε) phases both in ferromagnetic (FM) and nonmagnetic (NM) states is considered. Issues of fitting equation of states in the energy volume data points containing magnetism are discussed. Our results reveal two stable phase transition points, namely, from FM β-phase to FM a-phase at -28:2 GPa and from FM a-phase to NM β-phase at 123:7 GPa, which are consistent with previous experimental predictions. The meta-stable magnetic transformation points from FM to NM state for a, b, and e structures are 135:7 GPa, 84:0 GPa, and 168:7 GPa, respectively. Furthermore, both FM γ-phase and FM ε-phase undergo a meta-stable transformation to NM b-phase at 25:0 GPa and 42:3 GPa, respectively.