The conversion of molecular nitrogen to ammonia is a key biological and chemical process and represents one of the most challenging topics in chemistry and biology. In Nature the Mo-containing nitrogenase enzymes perform nitrogen 'fixation' via an iron molybdenum cofactor (FeMo-co) under ambient conditions. In contrast, industrially, the Haber-Bosch process reduces molecular nitrogen and hydrogen to ammonia with a heterogeneous iron catalyst under drastic conditions of temperature and pressure. This process accounts for the production of millions of tons of nitrogen compounds used for agricultural and industrial purposes, but the high temperature and pressure required result in a large energy loss, leading to several economic and environmental issues. During the last 40 years many attempts have been made to synthesize simple homogeneous catalysts that can activate dinitrogen under the same mild conditions of the nitrogenase enzymes. Several compounds, almost all containing transition metals, have been shown to bind and activate N₂ to various degrees. However, to date Mo(N₂)(HIPTN)₃N with (HIPTN)₃N= hexaisopropyl-terphenyl-triamidoamine is the only compound performing this process catalytically. In this review we describe how Density Functional Theory calculations have been of help in elucidating the reaction mechanisms of the inorganic compounds that activate or fix N₂. These studies provided important insights that rationalize and complement the experimental findings about the reaction mechanisms of known catalysts, predicting the reactivity of new potential catalysts and helping in tailoring new efficient catalytic compounds.