  This paper represents notes of the mini-courses given by the authors at the
GCGTA conference in Dortmund (2007), Ottawa-Saint Sauveur conference (2007),
Escola d'Algebra in Rio de Janeiro (2008) and Alagna (Italy, 2008) conference
on equations in groups. We explain here the Elimination process for solving
equations in a free group which has Makanin-Razborov process as a prototype. We
also explain how we use this process to obtain the structure theorem for
finitely generated fully residually free groups and many other results.
