  In this paper we study {\em terminal embeddings}, in which one is given a
finite metric $(X,d_X)$ (or a graph $G=(V,E)$) and a subset $K \subseteq X$ of
its points are designated as {\em terminals}. The objective is to embed the
metric into a normed space, while approximately preserving all distances among
pairs that contain a terminal. We devise such embeddings in various settings,
and conclude that even though we have to preserve $\approx|K|\cdot |X|$ pairs,
the distortion depends only on $|K|$, rather than on $|X|$.
  We also strengthen this notion, and consider embeddings that approximately
preserve the distances between {\em all} pairs, but provide improved distortion
for pairs containing a terminal. Surprisingly, we show that such embeddings
exist in many settings, and have optimal distortion bounds both with respect to
$X \times X$ and with respect to $K \times X$.
  Moreover, our embeddings have implications to the areas of Approximation and
Online Algorithms. In particular, [ALN08] devised an $\tilde{O}(\sqrt{\log
r})$-approximation algorithm for sparsest-cut instances with $r$ demands.
Building on their framework, we provide an $\tilde{O}(\sqrt{\log
|K|})$-approximation for sparsest-cut instances in which each demand is
incident on one of the vertices of $K$ (aka, terminals). Since $|K| \le r$, our
bound generalizes that of [ALN08].
