The main aim of this volume is to present a systematic theory of partial groupoids, the so-called "paragoids", such as with a single partial binary operation, giving the foundations of this theory, the main problems, and its most important results with full proofs. Attention is paid to specific features of the theory of partial groupoids. This theory is distinct from the theory of total operations (groups and semi-groups) and the theory of transformations, but they are connected, and their relations are also studied. This book should be of interest to researchers of general algebraic systems, group theory, functional analysis and information theory.