  Let $\mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal
$\mathfrak{M}$-universal if every $X\in\mathfrak{M}$ can be isometrically
embedded in $Y$ but there are no proper subsets of $Y$ satisfying this
property. We find conditions under which, for given metric space $X$, there is
a class $\mathfrak{M}$ of metric spaces such that $X$ is minimal
$\mathfrak{M}$-universal. We generalize the notion of minimal
$\mathfrak{M}$-universal metric space to notion of minimal
$\mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to
an isomorphism, for these classes. The necessary and sufficient conditions
under which the disjoint union of the metric spaces belonging to a class
$\mathfrak{M}$ is minimal $\mathfrak{M}$-universal are found. Examples of
minimal universal metric spaces are constructed for the classes of the
three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal
universal metric spaces are found for some subclasses of the class of metric
spaces $X$ which possesses the following property. Among every three distinct
points of $X$ there is one point lying between the other two points.
