  I introduce a new geometrical approach to thermo--statistical mechanics. Here
I highlight the main physical ideas, and how do they translate into geometrical
language. I contrast the present approach with previous
thermo--statistical--geometrical formalisms, (pseudo-)Riemannian [Weinhold
1975; Ruppeiner 1979] as well as Euclidean [Gilmore 1984; Gross& Votyakov 1999]
or Euclidean--looking [Levine 1986]. I point out the relevance of this approach
within the contexts of non--extensive statistical thermodynamics [Tsallis 1988,
2000]. I show how the language of Riemannian geometry can be used as a powerful
investigation tool,leading to several new and profound questions, and some
temptative answers, on thermo--statistical mechanics.
