  The quest for a theory of cities that could offer a quantitative and
systematic approach to manage cities is at the top priority, given the
challenges humanity faces due to the increasing urbanization and densification
of cities. If such a theory is feasible, then its formulation must be in a
mathematical way. As a contribution to organizing the mathematical ideas that
deal with such a systematic way of understanding urban phenomena, we present
this material, concentrating on one important aspect of what recently has been
called the new science of cities. In this paper, we review the main
mathematical models present in the literature that aim at explaining the origin
and emergence of urban scaling. We intend to present the models, identify
similarities and connections between them, and find situations in which
different models lead to the same output. In addition, we report situations in
which some ideas initially introduced in a particular model can also be
introduced in another model, generating more diversification and increasing the
scope of the models. The models treated in this paper explain urban scaling
from different premises: from gravity ideas, passing through densification
ideas and cites' geometry, to a hierarchical organization and socio-network
properties. We also investigate scenarios in which these different fundamental
ideas could be interpreted as similar -- where the similarity is likely but not
obvious. Furthermore, in what concerns the gravity idea, we propose a general
framework that includes all gravity models analyzed as a particular case.
