Noncommutative geometry in the sense of Connes

Khalkhali, Basic noncommutative geometry

Varilly, An introduction to noncommutative geometry

Landi, An introduction to noncommutative spaces and their geometries

Khalkhali and Marcolli, An invitation to noncommutative geometry

Gracia-Bondia et al, Elements of noncommutative geometry

Connes, Noncommutative differential geometry

Connes, Noncommutative geometry

Connes and Marcolli, Noncommutative geometry, quantum fields and motives

Background:

Douglas, Banach algebra techniques in operator theory

Higson and Roe, Analytic K-homology