This book provides deep insights into current research in the rapidly developing field of connexive logic and includes contributions from leading scholars in the field. Unlike the most well known systems of non-classical logic, systems of connexive logic are contra-classical in that they both reject certain classically valid principles and validate schemata that are not valid classically. The history of modern formal connexive logic may be seen to have started with Storrs McCall's dissertation "Non-classical Propositional Calculi" (Oxford) 1963, thus roughly 60 years ago. While at the turn of the 21st century connexive logic was a virtually dead research programme, the situation has changed significantly after the inclusion of an entry on connexive logic in the Stanford Encyclopedia of Philosophy in 2006 and the beginning of a series of annual workshops on connexive logic in 2015. Nowadays, connexive logic is a vibrant area, and the present volume offers an exciting glimpse on recent work in connexive logic. The topics range from discussions of the notion of connexivity in logic, to conditional logic, relevance logic, and experimental philosophy. The volume contains an introduction that puts the various chapters into perspective and concludes with a list of open problems in connexive logic. The volume will be of interest to logicians and philosophers interested in non-classical logic. Chapters 1, 9 and 10, are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.