Is There Anything Interesting to Say About Dimensions in Physics?

The theory of dimensions in physics is astonishingly rich. It can be viewed at different levels of abstraction and, at any of these levels, reveals deep suggestions. The relevant theory was initially developed to get a useful mean to reduce the number of variables in experiments. Within this context Rayleigh method and Buckingham Π theorem are highly conceptual working tools. Further elements of novelty have emerged during the last years and methods, directly or indirectly, linked to dimensional analysis, have become a central issue to treat families of differential equations, to enter deeply in the so-called scaling relations characterizing the phenomenology, not only in physics but in social science, economy, biology, and so on. This article is an effort aimed at providing a reasonably comprehensive account of the theory and the relevant practical outcomes, which spans over a large variety of topics including classical issues in hydrodynamics but also in general relativity and quantum mechanics as well.