%0 Journal Article %T Equations and fully residually free groups %A Olga Kharlampovich %A Alexei Myasnikov %J Mathematics %D 2009 %I arXiv %X 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. %U http://arxiv.org/abs/0904.4482v1