An Introduction to Noncommutative Spaces and Their Geometries

These notes arose from a series of introductory seminars on noncommutative geometry the author gave at the University of Trieste. They are mainly an introduction to Connes' noncommutative geometry and could serve as a 'first aid kit' before one ventures into the landscape of Connes' theory. Emphasis is placed on noncommutative spaces seen as concrete spaces. Examples of noncommutative spaces are provided by noncommutative lattices. The latter are the subject of work the author is doing in collaboration with others. These notes are also meant to be an introduction to these researches.