Fong has developed a compositional framework by the name of \emph{decorated cospans} that is well-suited for modeling open' networks and systems. In this framework, open networks are seen as the morphisms of a category and can be composed as such allowing larger open networks to be built up from smaller ones. Much work has already been done in this direction, and is where the story of this dissertation starts. During the process of my first work to promote Fong's theory of decorated cospan categories to decorated cospan \emph{bicategories}, it was noticed that the morphisms which made up these decorated cospan categories had a certain unexpected and stringent characterization to them. From this observation, one of the main frameworks of the present thesis, namely structured cospans', was conceived by John Baez. The main version of the Structured cospans framework utilizes double categories' which are similar in flavor to bicategories in that they have a 2-dimensional structure to them. Shortly thereafter, we decided to revisit Fong's decorated cospan machinery also from the perspective of double categories to improve the original decorated cospans framework. Much work in this thesis is built on the foundations of double categories, and while working with both the structured cospans and decorated cospans frameworks, a double category of open Markov processes was also built in which coarse-grainings' appear as 2-morphisms and are shown to be compatible with `black-boxing'.