This paper is a brief version of the monography [1]. The m-M Calculus deals with the so-called m-M functions, i.e. functions of the form $f: D \rightarrow R$ ( $D = [a_1, b_1] \times \ldots \times [a_n, b_n]$, where $n > 0$ is any integer and $a_i, b_i \in R$ ) subjected to the following supposition: For each n-dimensional segment $\Delta = [\alpha_1 , \beta_1] \times \ldots \times [\alpha_n , \beta_n] \subset D$ a pair of real numbers, denoted by $m(f)(\Delta)$, $M(f)(\Delta)$, satisfying the conditions \begin {equation} m(f)(\Delta) \leq M(f)(\Delta), \\ \\ (for all \Delta \subset D, X \in \Delta) \end{equation} \begin{equation} \lim_{diam \Delta \to 0}(M(f)(\Delta)-m(f)(\Delta))=0, \\ \\ (where diam \Delta : (\sum(\beta_i - \alpha_i)^2)^{1/2}) \end {equation} is effectively given. Such an ordered pair $<m(f), M(f)>$ of mappings m(f), M(f) (both mapping the set of all $\Delta \subset D$ into R) is called an m-M pair of the function f. We also say that m(f), M(f) are generalized minimum and maximum for f respectively. For instance, with only few exceptions all elementary functions are m-M functions (Lemma 1.2). The conditions (1) and (2) are taken as axioms of the m-M calculus. A logical analysis of these axioms is given here and, in addition to the other results , a series of equivalences is proved which enable us to express some relationships for m-M functions by means of the corresponding relationships for their m-M 'pair s (see (2. 2), (2 .5), (2 .6 ), (2 .7) ). There are many various applications of the m-M calculus, such as \begin{itemize} \item Solving systems of inequalities, systems of equations (Section 1) \item Finding n-dimensional integrals (Section 1, Example 1.5) \item Solving any problem expressed by a positive $leq$ formula (Section 2), among others \begin{enumerate} \item Problem of constrained optimization (Problem 2.2, Problem 2.3) \item Problem of unconstrained optimization (Problem 2.1) \item Problems from Interval Mathematics (Problem 2.4) \end{enumerate} \item Finding functions satisfying a given m-M condition (e.g, funclional condition, or difference equation, or differential equation, Section 3). \end{itemize} As it is well known, by the usual methods of numerical analysis , assuming certain convergence conditions , we approximately determine, step-by-step, one solution of the given problem. However, applying the methods of m-M calculus we approximately determine all solutions of the given problem, and we assume almost nothing about the convergence conditions . The solutions are, as a rule, sought in a prescribed n -dimensional segment D. If the given problem, e.g. a system of equations, has no solutions in D, then applying the method of m-M calculus this can be established at a certain finite step k. The basic methodological idea of the m-M calculus is : it gives a sufficient condition $Cond(\Delta)$ which ensures that an n-dimensional segment $\Delta$ does not contain any solution of the considered problem P. Applying repeatedly this criterion, we reject from the original n -segment D those "pieces" which do not contain solutions, so that in the limit case the remaining "pieces" form the set of all solutions of the problem P (if indeed there is a solution of P).