This thesis gathers three papers written by the author during PhD study at Lancaster University. In addition to these three papers, this thesis also contains two complementary sections. These two complementary sections are an introduction and a conclusion. The introduction discusses some recurring themes of the thesis and parts of the history leading to those results proven herein. The conclusion briefly comments on the outcomes of the research, its place in the current mathematical literature, and explores possibilities for further research.

A common theme of this thesis is the study of Maurer-Cartan elements and their moduli spaces, that is their study up to homotopy or gauge equivalence. Within the three papers cited above (and thus within this thesis), three different applications of Maurer-Cartan elements are demonstrated. The first application constructs certain moduli spaces as models for unbased disconnected rational topological spaces. The second application constructs certain moduli spaces as those governing formal algebraic deformation problems over, not necessarily local, commutative differential graded algebras. The third application uses the presentation of L-infinity algebras as solutions to the Maurer-Cartan equation in certain commutative differential graded algebras to construct minimal models of quantum L-infinity algebras. These quantum homotopy algebras arise as the `higher genus' versions of classical (cyclic) homotopy algebras.