This paper traces the history of Maturity Models and their impact on Process Improvement from the early work of Shewhart to their current usage with DevOps. The history of modern process improvement can be traced at least to Shewhart. From his foundational process contributions and those of other innovators a variety of methods and tools to aid in process quality advancement were developed. This paper begins by reviewing those early steps and then focuses on the emergence of Maturity Models in the 1970s with initial approach by Nolan. The broad adoption of Maturity Models that followed through the success of the CMM and then the CMMI approaches is detailed. This then leads to a general survey of additional models developed for such areas as IT Service Management, ITIL, Project Management, Agile Development, DevOps, CERT, and MDM among others. Finally, this paper discusses the application of these models in the support of process improvement and their limitations. Readers of this paper can expect to gain an appreciation for the origins of these models and surrounding methods as well as an ability to conduct comparative analysis of such models to aid in their selection and application. Keywords: Process Improvement, Process Engineering, Maturity Models, Capability Maturity Models, CMM, CMMI, ITSM, ITIL, Agile, DevOps, History of Science, History of Computing, Software Engineering, Quality.