This text presents a survey of basic elements of the modern mathematical theory of hysteresis memory, with the goal to offer a rigorous theoretical tool for engineers working in the development of high precision technologies on micro- and nanoscale. There is an increasing demand for faster and more robust computational methods based on reliable mathematical models, and in advanced applications, memory effects cannot be neglected without producing undesired discrepancy between physical reality and theoretical predictions. The theory is illustrated by a selection of topics related to its applications in engineering practice, including piezoelectric or magnetostrictive materials, plasticity, or fluid diffusion in porous materials. Our approach to hysteresis is based on the notion of variational inequality, which is, on the one hand, a mathematical expression of the physical principle of energy exchange between different components of a complex system with memory, and, on the other hand, allows to derive efficient estimates of possible modeling and computational errors.