\noindent Convex cones over a real unital algebra, which in addition are closed under inversion, may seem peculiar. However, Convex Invertible Cones (CICs) naturally appear in stability analysis of continuous-time physical systems. \vskip 0.2cm \noindent With this motivation, in this talk we explore examples of CICs over some algebras and establish interconnections among them. \vskip 0.2cm \noindent This indicates at the importance of the study of rational functions, of non-commuting variables, with certain positivity properties. \vskip 0.2cm \noindent This talk is based on an ongoing research for many years. Some of it in collaboration with Daniel Alpay, Chapman University, California, Nir Cohen, Natal, Brazil and the late Leiba Rodman, from the College of William and Mary, Virginia.