This book provides a lucid exposition of the connections between non‎-commutative geometry and the famous Riemann Hypothesis‎,‎ focusing on the theory of one‎-dimensional varieties over a finite field‎.‎ The reader will encounter many important aspects of the theory‎,‎ such as Bombieri‎'‎s proof of the Riemann Hypothesis for function fields‎,‎ along with an explanation of the connections with Nevanlinna theory and non‎-commutative geometry‎.‎ The connection with non‎-commutative geometry is given special attention‎,‎ with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space‎,‎ and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes‎.‎ The exposition is accessible at the graduate level and above‎,‎ and provides a wealth of motivation for further research in this area‎.‎