This paper reviews a theoretical program to understand the field of low energy nuclear reactions (LENR), initially called cold fusion, which was a puzzle of the last three decades. A series of concepts, which were raised and partly elaborated, are presented. The key concept is a three-body mechanism, where one of the nuclei serves as a catalyst for the reaction of the other two. In these reactions, a single fusion product or two outgoing nuclei are created, along with the recoiled catalytic nucleus. They carry off the nuclear energy released, and heat the surrounding materials by multiple collisions. Importantly, they produce high energy γ radiation with negligible probability. So, this three-body idea solves the two major riddles of LENR, that nuclear reactions can occur at ordinary temperatures and without emitting γ radiation. Cross sections and rates are calculated using standard quantum mechanics, and accepted nuclear and solid-state physics. Their dependence on relevant densities, and on the reacting and catalytic nuclei, are explicit. The theoretical ideas and the results of their elaboration are compared with diverse data from LENR experiments with considerable success.