  We render a thorough, physicist's account of the formulation of the Standard
Model (SM) of particle physics within the framework of noncommutative
differential geometry (NCG). We work in Minkowski spacetime rather than in
Euclidean space. We lay the stress on the physical ideas both underlying and
coming out of the noncommutative derivation of the SM, while we provide the
necessary mathematical tools. Postdiction of most of the main characteristics
of the SM is shown within the NCG framework. This framework, plus standard
renormalization technique at the one-loop level, suggest that the Higgs and top
masses should verify 1.3 m_top \lesssim m_H \lesssim 1.73 m_top.
