  We develop various Ehrenfeucht-Fra\"{\i}ss\'{e} games for distances between
metric structures. We study two forms of distances: pseudometrics stemming from
mapping spaces onto each other with some form of approximate isomorphism, and
metrics stemming from measuring the distances between two spaces isometrically
embedded into a third space. Using an infinitary version of Henson's positive
bounded logic with approximations, we form Scott sentences capturing fixed
distances to a given space. The Scott sentences of separable spaces are in
$\mathcal{L}_{\omega_1\omega}$ for 0-distances and in
$\mathcal{L}_{\omega_2\omega}$ for positive distances.
