The paper concerns metric regularity of a multifunction \(F: X\to Y\), its characterization in the complete metric space setting through the strong slope of De Giorgi, Marino, and Tosques, and a large sample of results among those which appeared during the last 25 years. In the Banach space setting, sufficient conditions for metric regularity involve abstract \(F\)-coderivatives \(D^*F(x,y):Y^*\to X^*\), whereas necessary conditions involve several concrete \(F\)-coderivatives such as the Fréchet, \(s\)-Hölder smooth, Dini, or Mordukhovich (limiting Fréchet) ones. Metric regularity is derived also from openness of certain \(F\)-tangent multifunctions \(T_F(x,y): X\to Y\). Parametric results conclude “equi”-metric regularity of a family \(F_p: X\to Y\) of multifunctions depending on a point \(p\) of a metric space \(P\). Reference is made to some metric regularity results which are based on the Brouwer fixed point theorem, or equivalently the invariance of the domain, but do not enter in the framework developed in the paper.