The recently discovered superconductor MgB₂ with T_c at 39 K has great potential in superconducting microelectronics. Thermodynamics studies with the calculation of phase diagrams (CALPHAD) modelling technique show that due to the high volatility of Mg, MgB₂ is only thermodynamically stable under fairly high Mg overpressures for likely in situ growth temperatures. This provides a helpful insight into the appropriate processing conditions for MgB₂ thin films, including the identification of the pressure–temperature region for adsorption-controlled growth. The initial MgB₂ thin films were made by pulsed laser deposition followed by in situ annealing. The cross-sectional transmission electron microscopy reveals a nanocrystalline mixture of textured MgO and MgB₂ with very small grain sizes. A zero-resistance transition temperature of 34 K and a zero-field critical current density of 1.3 × 10⁶ A cm⁻² were obtained. The qualities of these films are limited by the thermodynamic stability conditions, which favour deposition techniques that can maintain a high flux of Mg.