This thesis is mainly about Proof Theory. It can be thought of as Proof Theory in the sense of Hilbert, Gentzen, Schutte, Buchholz, Rathjen, and in general what could be called the German school, but it is also influenced by many other branches, of which the bibliography might give an idea. Intuitionism and other philosophical approaches to mathematics are also an important part of what is studied, but the Leitmotif of this thesis is Cut Elimination. The first part of the thesis is concerned with countable coded ω-models of Bar Induction. In this part we work from a reverse mathematics point of view. A study for an ordinal analysis of the theory of Bar Induction (BI) is carried out, and the equivalence between the statement that every set is contained in an co-model of this theory (BI) and the well-ordering principle VX[W0(3E) WO(6x)] which says that if X is a well-ordering, then so is its Bachmann-Howard relativisation, is proven. This is a new result as far as we know, and, we hope, an important one. In the second part of the thesis we shift our viewpoint and consider intuitionistic logic and intuitionistic geometric theories. We show that geometric derivability in classical infinitary logic implies derivability in intuitionistic infinitary logic. Again, our main tool is Cut Elimination. Next, we present investigations regarding minimal logic and classical logical principles, and give a precise classification of excluded middle, ex falso, and double negation elimination. Other themes and roads are possible and, the author feels, important, but time limitations as well as a sickly and utterly daft adherence to deadlines did not permit him to carry out these studies in full. It is quite shameful.