This thesis is based on the research results on the existence of compatible spanning circuits, which were obtained by the author with different collaborators between September 2016 and April 2020. Apart from the introductory chapter (i.e., Chapter 1), the thesis contains six closely related research chapters (i.e., Chapters 2-7). Chapters 2, 6, 7 and part of Chapter 3 are mainly based on the research that the author carried out while he was working as a PhD student at the Northwestern Polytechnical University in Xi'an, P.R. China. Chapter 4 is based on the research that was carried out while the author was visiting the Institute for Discrete Mathematics and Algebra, Technische Universität Bergakademie Freiberg in Freiberg, Germany. The other parts of the thesis are mainly based on the research results of the author at the University of Twente, The Netherlands. This thesis is devoted to developing sufficient conditions for the existence of specific compatible spanning circuits in edgecolored graphs, and to establishing polynomial-time algorithms for finding them. The papers associated with these six research chapters have been listed below and have been published in (or submitted to) scientific journals. Papers underlying this thesis [1] Compatible spanning circuits in edge-colored graphs, Discrete Mathematics,