Hyperbolic toral automorphisms (viz. the [cat map][1] and its generalizations) are the fundamental examples of Anosov diffeomorphisms, and their suspensions are the fundamental examples of Anosov flows. This is because they are "structurally stable", i.e. small perturbations preserve the Anosov property. I am actually not even aware of any other concrete examples of Anosov diffeomorphisms or flows.

  [1]: http://en.wikipedia.org/wiki/Arnold%2527s_cat_map