Writing a research monograph on a 'hot topic' such as 'uncertainty propagation' is a somewhat daunting undertaking. Nontheless, we decided to collect our own views, supported by our own research efforts over the past 12-15 years on a number of aspects of this topic and summarize these for the possible enlightenment it might provide (for us, our students and others). The research results discussed below are thus necessarily filled with a preponderance of references to our own research reports and papers. In numerous of the references below (given at the conclusion of each chapter), we refer to CRSC-TRXX-YY. These refer to early Technical Report versions of manuscripts which can be found on the Center for Research in Scientific Computation website at North Carolina State University where XX refers to the year, e.g., XX = 03 is 2003, XX = 99 is 1999, while the YY refers to the number of the report in that year. These can be found at and downloaded from http://www.ncsu.edu/crsc/reports.html where they are listed by year. Our presentation here has an intended audience from the community of investigators in applied mathematics interested in deterministic and/or stochastic models and their interactions as well as scientists in biology, medicine, engineering and physics interested in basic modeling and inverse problems, uncertainty in modeling, propagation of uncertainty, and statistical modeling. We owe great thanks to our former and current students, postdocs and colleagues for their patience in enduring lectures, questions, feedback, and some proofreading. Special thanks are due (in no particular order) to Zack Kenz, Keri Rehm, Dustin Kaparun, Jared Catenacci, Katie Link, Kris Rinnevatore, Kevin Flores-- Provided by publisher.