  There are two main aims of the paper. The first one is to extend the
criterion for the precompactness of sets in Banach function spaces to the
setting of quasi-Banach function spaces. The second one is to extend the
criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb
R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function
spaces. These criteria are applied to establish compact embeddings of abstract
Besov spaces into quasi-Banach function spaces. The results are illustrated on
embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq
\infty$, into Lorentz-type spaces.
