  We establish necessary and sufficient conditions guaranteeing compactness of
embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin
spaces, in the general context of quasi-metric-measure spaces. Although stated
in the setting of quasi-metric spaces, the main results in this article are
new, even in the metric setting. Moreover, by considering the more general
category of quasi-metric spaces we are able to obtain these characterizations
for optimal ranges of exponents that depend (quantitatively) on the geometric
makeup of the underlying space.
