This book presents a comprehensive study of inequality problems in mechanics and applications, focusing on convex and nonconvex energy functions. It is a result of seven years of seminars and courses given at various universities. The book is intended for a wide audience, including mathematicians and engineers. It is influenced by the work of G. Fichera, J. L. Lions, G. Maier, and J. J. Moreau, who developed the theory of inequality problems. The author also acknowledges the contributions of several colleagues and friends. The book is divided into three parts: Introductory Topics, Inequality Problems, and Numerical Applications. The first part provides a mathematical background, including functional analysis, convex analysis, and related topics. The second part discusses inequality problems in various areas of mechanics, including variational inequalities, superpotentials, friction problems, and problems in plasticity and viscoplasticity. The third part focuses on numerical applications, including the numerical treatment of static and dynamic inequality problems. The book emphasizes the difference between equality and inequality problems in mechanics. Inequality problems are those whose variational forms are inequalities, expressing the principle of virtual power in its inequality form. These problems are fundamentally different from equality problems, which have variational forms that are equalities. The book also discusses the mathematical study of unilateral boundary value problems and the numerical treatment of inequality problems. The book is written for a wide audience, including mathematicians and engineers. It is written in a way that is accessible to readers unfamiliar with functional analysis, who are interested in mechanics and applications. The book provides a detailed treatment of inequality problems in mechanics and applications, with a focus on convex and nonconvex energy functions. It is a valuable resource for researchers and practitioners in the field of mechanics and applications.This book presents a comprehensive study of inequality problems in mechanics and applications, focusing on convex and nonconvex energy functions. It is a result of seven years of seminars and courses given at various universities. The book is intended for a wide audience, including mathematicians and engineers. It is influenced by the work of G. Fichera, J. L. Lions, G. Maier, and J. J. Moreau, who developed the theory of inequality problems. The author also acknowledges the contributions of several colleagues and friends. The book is divided into three parts: Introductory Topics, Inequality Problems, and Numerical Applications. The first part provides a mathematical background, including functional analysis, convex analysis, and related topics. The second part discusses inequality problems in various areas of mechanics, including variational inequalities, superpotentials, friction problems, and problems in plasticity and viscoplasticity. The third part focuses on numerical applications, including the numerical treatment of static and dynamic inequality problems. The book emphasizes the difference between equality and inequality problems in mechanics. Inequality problems are those whose variational forms are inequalities, expressing the principle of virtual power in its inequality form. These problems are fundamentally different from equality problems, which have variational forms that are equalities. The book also discusses the mathematical study of unilateral boundary value problems and the numerical treatment of inequality problems. The book is written for a wide audience, including mathematicians and engineers. It is written in a way that is accessible to readers unfamiliar with functional analysis, who are interested in mechanics and applications. The book provides a detailed treatment of inequality problems in mechanics and applications, with a focus on convex and nonconvex energy functions. It is a valuable resource for researchers and practitioners in the field of mechanics and applications.