This paper gives a self-contained, comprehensive treatment of the theories of sequence spaces and measures of noncompactness, as well as a survey of some of the authors' recent research results in these fields. It contains subjects of lectures at the universities of Nis, Novi Sad and Belgrade, Giessen (Germany) and Irbid (Jordan), and talks given by the authors at various international conferences in the Czech Republic, Germany, Hungary, India, Italy, Jordan, Poland and Yugoslavia.