  In this thesis, we introduce Cartesian double categories, motivated by the
work of Carboni, Kelly, Walters, and Wood on Cartesian bicategories. Moving
from bicategories to the slightly more generalized notion of double categories
allows us to set the whole theory inside the welcoming 2-category of double
categories, and to overcome technical problems that were caused by working with
left adjoints inside a general bicategory. Cartesian double categories that are
also fibrant are of particular interest to us. After describing some important
properties of Cartesian and fibrant double categories, we give a
characterization of the double category of Spans as a Cartesian double
category. Lastly, we talk about profunctors and give a potential framework for
their characterization as Cartesian double categories.
