Universal constants and natural systems of units in a spacetime of arbitrary dimension.

We study the properties of fundamental physical constants using the threefold classification of dimensional constants proposed by J.-M. Levy-Leblond: constants of objects (masses etc.), constants of phenomena (coupling constants), and "universal constants" (such as $c$ and $\hbar$). We show that all of the known "natural" systems of units contain at least one non-universal constant. We discuss possible consequences of such non-universality, e.g. dependence of some of these systems on the number of spatial dimensions. In the search for a "fully universal" system of units, we propose a set of constants that consists of $c$, $\hbar$, and a length parameter, discuss its origins and the connection to the possible kinematic groups discovered by Levy-Leblond and Bacry. Finally, we make some comments about the interpretation of these constants.