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In this paper we estimate the Kuratowski and the Hausdorff measures ofnoncompactness of bounded subsets of spaces of vector-valued bounded functionsand of vector-valued bounded differentiable functions. To this end, we use aquantitative characteristic modeled on a new equiconti\-nuity-type concept andclassical quantitative characteristics related to pointwise relativecompactness. We obtain new regular measures of noncompactness in the spacestaken into consideration. The established inequalities reduce to preciseformulas in some classes of subsets. We derive Ascoli-Arzel\` a typecompactness criteria.

Regular measures of noncompactness and Ascoli-Arzel\` a type compactness criteria in spaces of vector-valued function