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We study the properties of fundamental physical constants using the threefoldclassification of dimensional constants proposed by J.-M. L{\'e}vy-Leblond:constants of objects (masses, etc.), constants of phenomena (couplingconstants), and "universal constants" (such as $c$ and $\hbar$). We show thatall of the known "natural" systems of units contain at least one non-universalconstant. We discuss the possible consequences of such non-universality, e.g.,the dependence of some of these systems on the number of spatial dimensions. Inthe search for a "fully universal" system of units, we propose a set ofconstants that consists of $c$, $\hbar$, and a length parameter and discuss itsorigins and the connection to the possible kinematic groups discovered byL{\'e}vy-Leblond and Bacry. Finally, we give some comments about theinterpretation of these constants.

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