Starting with simple examples showing the relevance of cutting and pasting logics, the monograph develops a mathematical theory of combining and decomposing logics, ranging from propositional and first-order based logics to higher-order based logics as well as to non-truth functional logics. The theory covers mechanisms for combining semantic structures and deductive systems either of the same or different nature (for instance, two Hilbert calculi or a Hilbert calculus and a tableau calculus). The important issue of preservation of properties is extensively addressed. For instance, sufficient conditions are provided for a combined logic to be sound and complete when the original component logics are known to be sound and complete. The book brings the reader to the front line of current research in the field by showing both recent achievements and directions of future investigations (in particular, multiple open problems). It also provides examples of potential applications in emergent fields like security protocols, quantum computing, networks and argumentation theory, besides discussing more classical applications like software specification, knowledge representation, computational linguistics and modular automated reasoning. This monograph will be of interest to researchers and graduate students in mathematical logic, theory of computation and philosophical logic with no previous knowledge of the subject of combining and decomposing logics, but with a working knowledge of first-order logic. The book will also be relevant for people involved in research projects where logic is used as a tool and the need for working with several logics at the same time is mandatory (for instance, temporal, epistemic and probabilistic logics).