Let \(X\) and \(Y\) be complete metric spaces. The closed-set-valued mapping \(F: X\Rightarrow Y\) is called \textit{metrically regular} on the set \(V\subset X\times Y\) if there exists a number \(K>0\) such that \[ d(x, F^{-1}(y))\leq K\cdot d(y, F(x))\quad \forall(x,y)\in V. \] This notion is closely connected with important principles of smooth and nonsmooth analysis. So the classical Banach-Schauder open mapping theorem points out that a bounded linear operator \(A: X\to Y\) between Banach spaces is (globally) metrically regular if \(\text{Im}(A)= Y\). Also the well-known tangent space theorem of Ljusternik and the surjection theorem of Graves give a sufficient condition for the metric regularity of a smooth operator \(F:X\to Y\) near a point \((x^0,y^0)\) using the condition \(\text{Im}(F'(x^0))= Y\). In the paper the author gives a substantial survey about the theory of metric regularity of set-valued mappings and its connections with subdifferential calculus. It makes clear that the phenomenon behind the results about metric regularity are of metric origin, not connected with any linear structure of the spaces. In the first chapter, the author gives some necessary and sufficient conditions for the metric regularity. A central point is the equivalence of metric regularity to a certain surjection (openness or cover) property for the mapping \(F\) and to a generalized Lipschitz property for the mapping \(F^{-1}\). The second chapter is devoted to the theory of subdifferentials. The author summarizes basic theorems of the subdifferential calculus for different kinds of general subdifferentials (Fréchet subdifferential, \(\beta\)-subdifferentials with regard to a bornology \(\beta\), limiting subdifferentials, approximate subdifferential, Clarke subdifferential, axiomatic subdifferential) and fundamental results about normal cones of sets and coderivatives of set-valued mappings. Finally, the third chapter points out the close connection between subdifferential calculus and metric regularity. On the one hand, results of differential calculus are used for the characterization of local metric regularity. On the other hand, the main theorems of subdifferential calculus require additional regularity assumptions which can be described by metric regularity conditions of certain set-valued maps.