This book presents a history of the Midwestern Universities Research Association (MURA) during its lifetime from the early 1950s to the late 1960s. MURA was responsible for a number of important contributions to the science of particle accelerators, including the invention of fixed field alternating gradient accelerators (FFAG), as well as contributions to accelerator orbit theory, radio frequency acceleration techniques, colliding beams technology, orbit instabilities, computation methods, and designs of accelerator magnets and linear accelerator cavities. A number of students were trained by MURA in accelerator techniques, and went on to important posts where they made further contributions to the field. The authors were all members of the MURA staff and themselves made many contributions to the field. No other such history exists, and there are relatively few publications devoted to the history of particle accelerators.