The focus of the present paper is on ab initio approaches to the elasticity of materials. First, the methodology to determine single-crystalline elastic constants and polycrystalline moduli of ordered compounds as well as disordered alloys is introduced. In the second part, the methodology is applied on Fe, with a main focus on investigating the influence of magnetism on its elasticity and phase stability and simulating extreme loading conditions that go up to the theoretical tensile strength limits and beyond.