The issues of safety-risk are ubiquitous in our complex industrial communities around the globe. Accidents provoke re-thinking and review of a system's life cycle comprised of site, design, procedures, emergency preparation, and training of personnel. The fundamentals and viewpoints of several mathematical formulations of natural processes or of estimating the consequences conditional on an accident are emphasized, as well as siting analysis for a hazardous facility. These investigations are derived from experience and teaching models of predicting the transport of releases to the environment as part of a larger assessment of their impacts. Several mathematical extensions of calculus are explored for modeling relevant dynamics. Technology strives for a policy of prevention, of defense in depth, and of "let's get it right the first time"; however, accidents are perceived to occur too often affecting the lives and properties of many people. The intent was to include a perspective of why the nuclear energy option was chosen by many countries including information on energy equivalents of fuels, as well as the "Atoms for Peace" speech of President Eisenhower. The reality of accidents challenges societies to invest in design improvements, monitoring, and procedures to sustain the benefits these technologies offer.