By an ab initio density functional approach the structural and phase stability, electronic and magnetic properties, elastic constants, phonon dispersion, and defect formation of the Laves-phase compound ZrMn₂ for the C15, C14, and C36 crystal structures were investigated. In order to study the stability of magnetic phases, nonmagnetic and spin polarized calculations for ferro- and antiferromagnetic orderings were performed. At low temperatures, the ferromagnetic cubic C15 phase was obtained as the ground state, with the ferromagnetic hexagonal C14 and C36 phases being almost degenerate in energy. From the calculated temperature-dependent free energies a structural transformation from C15 to C14 at about Ttr = 200 K is predicted, confirming the experimentally observed C14 structure at elevated temperatures. Elastic properties were investigated for the nonmagnetic and ferromagnetic C14 and C15 phases. Structural stability studies based on the calculated temperature-dependent free vibrational energies very strongly favor the existence of ferromagnetic phases. Point defect formation properties for vacancies and antiste defects were calculated by combining a supercell approach with a statistical mechanics model. Mn antisites are the most favorable defects broadening the homogeneity range of the ZrMn₂ phase toward the Mn-rich side. The existence of ordered Mn-rich compounds is predicted. Large magnetic moment of 3 μB for Mn antisite defects are derived.