A metric space $\text{M=}\left( M;\text{d} \right)$ is homogeneous if for every isometry $f$ of a finite subspace of $\text{M}$ to a subspace of $\text{M}$ there exists an isometry of $\text{M}$ onto $\text{M}$ extending $f$. The space $\text{M}$ is universal if it isometrically embeds every finite metric space $\text{F}$ with $\text{dist}\left( \text{F} \right)\subseteq \text{dist}\left( \text{M} \right)$ (with $\text{dist}\left( \text{M} \right)$ being the set of distances between points in $\text{M}$).

A metric space $U$ is a Urysohn metric space if it is homogeneous, universal, separable, and complete. (We deduce as a corollary that a Urysohn metric space $U$ isometrically embeds every separable metric space $\text{M}$ with $\text{dist}\left( \text{M} \right)\subseteq \text{dist}\left( U \right)$.)

The main results are: (1) A characterization of the sets $\text{dist}\left( U \right)$ for Urysohn metric spaces $U$. (2) If $R$ is the distance set of a Urysohn metric space and $\text{M}$ and $\text{N}$ are two metric spaces, of any cardinality with distances in $R$, then they amalgamate disjointly to a metric space with distances in $R$. (3) The completion of every homogeneous, universal, separable metric space $\text{M}$ is homogeneous.