The adoption of a rigorous software process is well known to represent a key factor for the quality of the software product and the most effective usage of the human resources available to a software project. The Unified Process, in particular its commercial packaging known as the RUP (Rational Unified Process) has been one of the most widely used software process models in the software industry for a number of years. We present the application of the Unified Process and of the RUP to a variety of Monte Carlo simulation projects in High Energy Physics, space science and medical physics environments. We illustrate how the UP/RUP provide a flexible process framework, that can be tailored to the different needs of individual software projects. We describe the experience with different approaches (top-down and bottom-up) to the implementation of the process in software organisations. We document a critical analysis of the effects of the adoption of the UP/RUP, and discuss the relative benefits of the public (UP) and commercial (RUP) versions of the process.