  The aim of the paper is to characterize (pre)compactness in the spaces of
Lipschitz/H\"older continuous mappings acting from a compact metric space to a
normed space. To this end some extensions and generalizations of already
existing compactness criteria for the spaces of bounded and continuous mappings
with values in normed spaces needed to be established. Those auxiliary results,
which are interesting on their own since they use a concept of equicontinuity
not seen in the literature, are based on an abstract compactness criterion
related to the recently introduced notion of an equinormed set.
