  This is an edited write-up of lecture notes of the 7-th Appalachian set
theory workshop of the same title led by the first named author at the Cornell
University on November 22, 2008. A draft version of the notes was prepared by
the second named author. This presentation is largely complementary to the
earlier survey by the first-named author (Hyperlinear and sofic groups: a brief
guide, Bull. Symb. Logic 14 (2008), pp. 449-480; arXiv:0804.3968v8 [math.GR]).
