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 beautiful but bewildering landscape of Connes' theory. The main difference with other available introductions to Connes' work is the emphasis on noncommutative spaces seen as concrete spaces. Important examples of noncommutative spaces are provided by noncommutative lattices. The latter are subject of intensive work the author is doing in collaboration with others. These notes are also meant to be an introduction to these researches.