Presenting the most important results of a new branch of functional analysis - subdifferential calculus and its applications - this monograph details new tools and techniques of convex and non-smooth analysis, such as Kantorovich spaces, vector duality, Boolean-valued and infinitesimal versions of non-standard analysis, covering a wide range of topics. The book aims to fill the gap between the theoretical core of modern functional analysis and its applicable sections, such as optimization, optimal control, mathematical programming, economics and related subjects. The material is intended for theoretical mathematicians looking for possible new applications, and applied mathematicians seeking powerful contemporary theoretical methods.